38:
6231:
114:
6293:
for 11). It relies on the fact that, given the sequence already generated, both of those sources are still exchangeable sequences of bits, and thus eligible for another round of extraction. While such generation of additional sequences can be iterated infinitely to extract all available entropy, an infinite amount of computational resources is required, therefore the number of iterations is typically fixed to a low value â this value either fixed in advance, or calculated at runtime.
4447:
6542:), so from 14 bits of input 10 bits of output were generated, as opposed to 3 bits through the von Neumann algorithm alone. The constant output of exactly 2 bits per round per bit pair (compared with a variable none to 1 bit in classical VN) also allows for constant-time implementations which are resistant to
6292:
The iterated version of the von
Neumann algorithm, also known as advanced multi-level strategy (AMLS), was introduced by Yuval Peres in 1992. It works recursively, recycling "wasted randomness" from two sources: the sequence of discard-non-discard, and the values of discarded pairs (0 for 00, and 1
3051:
fraction of the time, and that this corresponds exactly with the peak of the
Gaussian. The asymptotic equipartition property essentially states that this peak is infinitely sharp, with infinite fall-off on either side. That is, given the set of all possible infinitely long strings of
6557:
Another tweak was presented in 2016, based on the observation that the
Sequence2 channel doesn't provide much throughput, and a hardware implementation with a finite number of levels can benefit from discarding it earlier in exchange for processing more levels of Sequence1.
5398:
5767:
5114:
3011:
3335:
are, in a certain sense, "stronger" than the
Bernoulli process, which is merely ergodic but not mixing. However, such processes do not consist of independent random variables: indeed, many purely deterministic, non-random systems can be mixing).
6288:
This decrease in efficiency, or waste of randomness present in the input stream, can be mitigated by iterating the algorithm over the input data. This way the output can be made to be "arbitrarily close to the entropy bound".
4212:
1778:
3858:
5940:
4833:
3531:
3331:; in a certain sense, it is the single most random process possible; nothing is 'more' random than the Bernoulli process (although one must be careful with this informal statement; certainly, systems that are
2321:
6188:
The von
Neumann extractor uses two input bits to produce either zero or one output bits, so the output is shorter than the input by a factor of at least 2. On average the computation discards proportion
4625:
5483:
934:
4284:
2757:
2623:
5264:
5222:
6084:
Represent the observed process as a sequence of zeroes and ones, or bits, and group that input stream in non-overlapping pairs of successive bits, such as (11)(00)(10)... . Then for each pair,
3288:
4076:
4017:
3724:
3686:
3060:
occurring in the
Bernoulli process, this set is partitioned into two: those strings that occur with probability 1, and those that occur with probability 0. This partitioning is known as the
3979:
3893:
2439:
3630:
2050:
6864:
5596:
2821:
2135:
1225:
1172:
455:
4926:
3564:
479:. Much of what can be said about the Bernoulli process can also be generalized to more than two outcomes (such as the process for a six-sided die); this generalization is known as the
4399:
3419:
1045:
969:
4990:
5531:
4491:
1604:
4884:
4545:
6381:(If the length of the input is odd, the last bit is completely discarded.) Then the algorithm is applied recursively to each of the two new sequences, until the input is empty.
5148:
4432:
4110:
3944:
7399:
6044:
5987:
1375:
2912:
4680:
1310:
1069:
3766:
1974:
1921:
1486:
6009:
3194:
2900:
2082:
405:
6534:
Starting from step 1, the input is a concatenation of sequence 2 and sequence 1 from the previous step (the order is arbitrary but should be fixed). The final output is
820:
6296:
More concretely, on an input sequence, the algorithm consumes the input bits in pairs, generating output together with two new sequences, () gives AMLS paper notation:
1115:
872:
7223:
3156:
3130:
1864:
781:
as a random sequence of independent realisations of a random variable that can take values of heads or tails. The state space for an individual value is denoted by
5804:
5252:
3100:
2874:
2469:
1808:
4982:
7826:
6819:
4310:
7356:
7336:
5824:
5588:
4953:
4724:
4700:
4648:
4330:
3913:
2353:
2237:
2217:
2197:
2177:
1994:
1941:
1888:
1448:
7740:
5953:
represents a sequence of coin flips, then the associated
Bernoulli sequence is the list of natural numbers or time-points for which the coin toss outcome is
567:
615:
If the process is infinite, then from any point the future trials constitute a
Bernoulli process identical to the whole process, the fresh-start property.
4119:
7657:
1635:
639:
Two other common interpretations of the values are true or false and yes or no. Under any interpretation of the two values, the individual variables
7667:
7341:
7351:
7709:
3039:
The combination of the law of large numbers, together with the central limit theorem, leads to an interesting and perhaps surprising result: the
7424:
7606:
2088:. Nevertheless, one can still say that some classes of infinite sequences of coin flips are far more likely than others, this is given by the
398:
6750:
703:
One experiment with only two possible outcomes, often referred to as "success" and "failure", usually encoded as 1 and 0, can be modeled as a
7896:
7886:
7409:
6612:
7796:
7760:
3774:
632:
are often called "success" and "failure". Thus, when expressed as a number 0 or 1, the outcome may be called the number of successes on the
7713:
5867:
4729:
999:
stands for tails), with the rest of (infinitely long) sequence taken as "don't care". These sets of finite sequences are referred to as
8064:
7801:
5827:
3430:
3353:
2246:
587:
4352:, and in this case, it is 1. The associated eigenvector is the invariant measure: in this case, it is the Bernoulli measure. That is,
6911:
6812:
4553:
5414:
877:
7866:
7444:
7414:
6779:
6653:
6587:
6275:
3040:
2089:
391:
379:
338:
81:
59:
6708:
5555:. Informally, this is exactly what it sounds like: just "add one" to the first position, and let the odometer "roll over" by using
4217:
2646:
680:, .... That passage of time and the associated notions of "past" and "future" are not necessary, however. Most generally, any
7717:
7701:
7911:
7616:
6836:
3320:
2527:
439:
5559:
as the odometer rolls over. This is nothing more than base-two addition on the set of infinite strings. Since addition forms a
7816:
7781:
7750:
7745:
7384:
7181:
7098:
487:
269:
205:
7755:
7083:
6205: = 1/2 for the original process (in which case the output stream is 1/4 the length of the input stream on average).
5853:
Suppose a
Bernoulli process formally defined as a single random variable (see preceding section). For every infinite sequence
5393:{\displaystyle {\frac {1}{2}}B_{n}\left({\frac {y}{2}}\right)+{\frac {1}{2}}B_{n}\left({\frac {y+1}{2}}\right)=2^{-n}B_{n}(y)}
7379:
7186:
766:
739:
317:
178:
7105:
7841:
7721:
8090:
8069:
7846:
7682:
7581:
7566:
6978:
6894:
6805:
6182:
6161:
In the output stream 0 and 1 are equally likely, as 10 and 01 are equally likely in the original, both having probability
7856:
7492:
6553:
if (Bit1 â Bit2) { output(1, Sequence1) output(Bit1) } else { output(0, Sequence1) output(Bit1, Sequence2) }
5160:
4349:
7851:
3324:
3202:
7454:
4025:
7038:
6983:
6899:
6648:
Dean J. Driebe, Fully
Chaotic Maps and Broken Time Symmetry, (1999) Kluwer Academic Publishers, Dordrecht Netherlands
3984:
3691:
3653:
486:
The problem of determining the process, given only a limited sample of Bernoulli trials, may be called the problem of
7786:
7776:
7419:
7389:
3949:
3863:
2903:
2361:
7791:
6956:
6854:
5762:{\displaystyle T\left(1,\dots ,1,0,X_{k+1},X_{k+2},\dots \right)=\left(0,\dots ,0,1,X_{k+1},X_{k+2},\dots \right).}
3572:
2003:
Note that the probability of any specific, infinitely long sequence of coin flips is exactly zero; this is because
173:
7502:
7078:
6859:
7871:
7672:
7586:
7571:
6961:
5847:
2491:
2006:
1629:
appears in the sequence. There are several different kinds of notations for the above; a common one is to write
1247:
980:
289:
7705:
7591:
7013:
5109:{\displaystyle \left(y)={\frac {1}{2}}f\left({\frac {y}{2}}\right)+{\frac {1}{2}}f\left({\frac {y+1}{2}}\right)}
3067:
The size of this set is interesting, also, and can be explicitly determined: the logarithm of it is exactly the
2765:
2098:
1177:
1124:
7093:
7068:
6792:
4893:
3539:
1228:
708:
503:
348:
343:
232:
217:
52:
46:
7811:
7394:
6929:
4355:
3389:
1014:
939:
8006:
7996:
7687:
7469:
7208:
7073:
6884:
5495:
4457:
327:
198:
7291:
7948:
7876:
7135:
6253:
1494:
754:
704:
476:
222:
63:
4838:
4504:
7971:
7953:
7933:
7928:
7647:
7479:
7459:
7306:
7249:
7088:
6998:
6069:
5255:
5122:
4887:
4406:
4084:
3918:
3377:
3033:
3006:{\displaystyle n!={\sqrt {2\pi n}}\;n^{n}e^{-n}\left(1+{\mathcal {O}}\left({\frac {1}{n}}\right)\right)}
2824:
2154:
2150:
1118:
720:
322:
227:
193:
7439:
363:
6741:. 2016 IEEE International Symposium on Hardware Oriented Security and Trust (HOST). Maclean, VA, USA.
6020:
5963:
5850:
of a Bernoulli process. However, the term has an entirely different formal definition as given below.
1315:
8046:
8001:
7991:
7732:
7677:
7652:
7621:
7601:
7361:
7346:
7213:
6073:
6059:
5563:, and the Bernoulli process was already given a topology, above, this provides a simple example of a
4435:
3332:
2519:
2240:
2146:
353:
247:
140:
4653:
1256:
1050:
113:
8041:
7881:
7806:
7611:
7371:
7281:
7171:
6733:
6150:. Then, according to the table above, these pairs are translated into the output of the procedure:
5831:
5560:
3735:
3727:
3068:
3029:
1946:
1893:
1453:
696:
in the process are simply two from a set of random variables indexed by {1, 2, ...,
312:
254:
242:
237:
6177:. This extraction of uniform randomness does not require the input trials to be independent, only
5992:
3173:
2879:
2055:
8011:
7976:
7891:
7861:
7631:
7626:
7449:
7286:
6951:
6889:
6828:
3328:
3061:
2327:
almost certainly, that is, the events which do not satisfy this limit have zero probability. The
830:
655:
In many applications time passes between trials, as the index i increases. In effect, the trials
608:
is unknown, however, the past informs about the future indirectly, through inferences about
299:
188:
128:
105:
7692:
3536:
The Bernoulli measure, defined above, is translation-invariant; that is, given any cylinder set
3315:
posed a question about the Bernoulli process regarding the possibility of a given process being
784:
2095:
To conclude the formal definition, a Bernoulli process is then given by the probability triple
1082:
839:
8031:
7836:
7487:
7244:
7161:
7130:
7023:
7003:
6993:
6849:
6775:
6649:
6608:
6583:
5564:
5151:
4333:
3644:
3636:
3294:
2328:
1997:
778:
358:
163:
7697:
7434:
3135:
8051:
7938:
7821:
7191:
7166:
7115:
7043:
6966:
6919:
6742:
6681:
6047:
4494:
3373:
3345:
3327:. This breakthrough resulted in the understanding that the Bernoulli process is unique and
3312:
3105:
1817:
976:
480:
183:
5775:
5230:
3078:
2852:
2447:
1786:
8016:
7916:
7901:
7662:
7596:
7274:
7218:
7201:
6946:
5552:
5546:
5489:
4958:
4113:
3361:
2472:
1867:
1811:
649:
590:
506:
472:
435:
431:
259:
210:
7831:
7063:
4289:
6701:
8021:
7986:
7906:
7512:
7259:
7176:
7145:
7140:
7120:
7110:
7053:
7048:
7028:
7008:
6973:
6941:
6924:
6241:
5809:
5573:
4938:
4929:
4709:
4685:
4633:
4315:
3898:
3422:
3349:
3323:. The question long defied analysis, but was finally and completely answered with the
2338:
2324:
2222:
2202:
2182:
2162:
1979:
1926:
1873:
979:. The sets in this topology are finite sequences of coin flips, that is, finite-length
833:
597:
274:
1391:
8084:
7923:
7464:
7301:
7296:
7254:
7196:
7018:
6934:
6874:
6543:
4345:
1008:
1004:
459:
147:
7981:
7943:
7497:
7429:
7318:
7313:
7125:
7058:
7033:
6869:
6178:
4703:
4207:{\displaystyle {\mathcal {L}}_{T}(f+g)={\mathcal {L}}_{T}(f)+{\mathcal {L}}_{T}(g)}
3640:
2085:
1000:
552:
374:
284:
168:
7561:
17:
3071:
of the Bernoulli process. Once again, consider the set of all strings of length
462:, possibly with an unfair coin (but with consistent unfairness). Every variable
8026:
7545:
7540:
7535:
7525:
7328:
7269:
7264:
7228:
6988:
6879:
6732:
RoĆŸiÄ, Vladimir; Yang, Bohan; Dehaene, Wim; Verbauwhede, Ingrid (3â5 May 2016).
6245:
4446:
3383:
3357:
419:
294:
135:
123:
3043:. Put informally, one notes that, yes, over many coin flips, one will observe
1773:{\displaystyle P(X_{1}=x_{1},X_{2}=x_{2},\cdots ,X_{n}=x_{n})=p^{k}(1-p)^{n-k}}
700:}, the finite cases, or by {1, 2, 3, ...}, the infinite cases.
8036:
7576:
7520:
7404:
6209:
6014:
5534:
4501:
The above can be made more precise. Given an infinite string of binary digits
3316:
423:
152:
98:
6746:
6686:
6669:
7530:
5556:
6185:
of bits: all sequences that are finite rearrangements are equally likely.
3382:
One way to create a dynamical system out of the Bernoulli process is as a
604:
is known, past outcomes provide no information about future outcomes. (If
5858:
4341:
3158:. By using Stirling's approximation, putting it into the expression for
1079:
If the chances of flipping heads or tails are given by the probabilities
972:
279:
2876:
for a sufficiently long sequences of coin flips, that is, for the limit
2084:. A probability equal to 1 implies that any given infinite sequence has
1388:
Given a cylinder set, that is, a specific sequence of coin flip results
7357:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
6797:
5772:
It leaves the Bernoulli measure invariant only for the special case of
3102:. Of these, only a certain subset are likely; the size of this set is
6578:
Dekking, F. M.; Kraaikamp, C.; LopuhaÀ, H. P.; Meester, L. E. (2005).
3853:{\displaystyle \left(f\circ T^{-1}\right)(\sigma )=f(T^{-1}(\sigma ))}
3170:), solving for the location and width of the peak, and finally taking
4438:! This coincidence of naming was presumably not known to Bernoulli.
2141:
Law of large numbers, binomial distribution and central limit theorem
1488:, the probability of observing this particular sequence is given by
442:
that takes only two values, canonically 0 and 1. The component
6064:
From any Bernoulli process one may derive a Bernoulli process with
5935:{\displaystyle \mathbb {Z} ^{x}=\{n\in \mathbb {Z} :X_{n}(x)=1\}\,}
4828:{\displaystyle T(b_{0},b_{1},b_{2},\cdots )=(b_{1},b_{2},\cdots ),}
4434:
to act on polynomials, then the eigenfunctions are (curiously) the
1071:
are the finite-length sequences of coin flips (the cylinder sets).
6735:
Iterating Von Neumann's post-processing under hardware constraints
4445:
3526:{\displaystyle T(X_{0},X_{1},X_{2},\cdots )=(X_{1},X_{2},\cdots )}
2316:{\displaystyle {\bar {X}}_{n}:={\frac {1}{n}}\sum _{i=1}^{n}X_{i}}
711:
beside the Bernoullis may be derived from the Bernoulli process:
4620:{\displaystyle y=\sum _{n=0}^{\infty }{\frac {b_{n}}{2^{n+1}}}.}
676:, ... happen at "points in time" 1, 2, ...,
6801:
5478:{\displaystyle y=\sum _{n=0}^{\infty }{\frac {b_{n}}{3^{n+1}}}}
3386:. There is a natural translation symmetry on the product space
929:{\displaystyle \Omega =2^{\mathbb {N} }=\{H,T\}^{\mathbb {N} }}
6793:
Using a binary tree diagram for describing a Bernoulli process
6670:"Iterating Von Neumann's Procedure for Extracting Random Bits"
6224:
5989:
is also a random subset of the index set, the natural numbers
5150:
to functions that are on polynomials, one finds that it has a
2490:
successive coin flips, that is, given the set of all possible
2478:
One is often interested in knowing how often one will observe
1003:
in the product topology. The set of all such strings forms a
753:
The number of failures needed to get one success, which has a
31:
6631:-adic one-dimensional maps and the Euler summation formula",
4864:
4474:
4348:
and corresponding eigenvalues. The largest eigenvalue is the
4279:{\displaystyle {\mathcal {L}}_{T}(af)=a{\mathcal {L}}_{T}(f)}
2752:{\displaystyle \mathbb {P} ()={n \choose k}p^{k}(1-p)^{n-k},}
765:
The negative binomial variables may be interpreted as random
5492:, as conventionally defined. This is one reason why the set
5167:
5129:
5002:
4413:
4362:
4256:
4224:
4184:
4158:
4126:
4091:
4032:
3996:
3955:
3925:
3869:
3703:
3665:
3551:
3301:
stands for entropy; not to be confused with the same symbol
2975:
2113:
1056:
1029:
2618:{\displaystyle N(k,n)={n \choose k}={\frac {n!}{k!(n-k)!}}}
777:
The Bernoulli process can be formalized in the language of
7337:
Autoregressive conditional heteroskedasticity (ARCH) model
2632:, then the total probability of seeing a string of length
5551:
Another way to create a dynamical system is to define an
6865:
Independent and identically distributed random variables
6549:
Von NeumannâPeres (iterated) main operation pseudocode:
6197:) of the input pairs(00 and 11), which is near one when
6257:
5949:
associated with the Bernoulli process. For example, if
2849:
Of particular interest is the question of the value of
2830:
As we can see from the above formula that, if n=1, the
2823:. The probability measure thus defined is known as the
761:), a special case of the negative binomial distribution
596:
Independence of the trials implies that the process is
7342:
Autoregressive integrated moving average (ARIMA) model
5258:. Indeed, the Bernoulli polynomials obey the identity
2486:
coin flips. This is given by simply counting: Given
555:
6023:
5995:
5966:
5870:
5812:
5778:
5599:
5576:
5498:
5417:
5267:
5233:
5163:
5125:
4993:
4961:
4941:
4896:
4841:
4732:
4712:
4688:
4656:
4636:
4556:
4507:
4460:
4409:
4358:
4318:
4292:
4220:
4122:
4087:
4028:
3987:
3952:
3921:
3901:
3866:
3777:
3738:
3694:
3656:
3575:
3542:
3433:
3392:
3344:
The Bernoulli process can also be understood to be a
3205:
3176:
3138:
3108:
3081:
2915:
2882:
2855:
2768:
2649:
2530:
2450:
2364:
2341:
2249:
2225:
2205:
2185:
2165:
2101:
2058:
2009:
1982:
1949:
1929:
1896:
1876:
1820:
1789:
1638:
1497:
1456:
1394:
1318:
1259:
1180:
1127:
1085:
1053:
1017:
942:
880:
842:
787:
586:
In other words, a Bernoulli process is a sequence of
3356:, in one of several different ways. One way is as a
874:. It is common to examine either the one-sided set
7964:
7769:
7731:
7640:
7554:
7511:
7478:
7370:
7327:
7237:
7154:
6910:
6835:
6580:
A modern introduction to probability and statistics
5217:{\displaystyle {\mathcal {L}}_{T}B_{n}=2^{-n}B_{n}}
1227:for the two-sided process). In another word, if a
29:
Random process of binary (boolean) random variables
6772:Probability and Stochastic Processes for Engineers
6388:using 1 for H and 0 for T, is processed this way:
6201:is near zero or one, and is minimized at 1/4 when
6038:
6003:
5981:
5934:
5818:
5798:
5761:
5582:
5525:
5477:
5392:
5246:
5216:
5142:
5108:
4976:
4947:
4920:
4878:
4827:
4718:
4694:
4674:
4642:
4619:
4539:
4485:
4426:
4393:
4324:
4304:
4278:
4206:
4104:
4070:
4011:
3973:
3938:
3907:
3887:
3852:
3760:
3718:
3680:
3624:
3558:
3525:
3413:
3283:{\displaystyle H=-p\log _{2}p-(1-p)\log _{2}(1-p)}
3282:
3188:
3150:
3124:
3094:
3005:
2894:
2868:
2815:
2751:
2617:
2463:
2433:
2347:
2315:
2231:
2211:
2191:
2171:
2129:
2076:
2044:
1988:
1968:
1935:
1915:
1882:
1858:
1802:
1772:
1598:
1480:
1442:
1369:
1304:
1219:
1166:
1109:
1063:
1039:
963:
928:
866:
814:
561:
458:. Prosaically, a Bernoulli process is a repeated
4071:{\displaystyle {\mathcal {L}}_{T}f=f\circ T^{-1}}
2702:
2689:
2628:If the probability of flipping heads is given by
2568:
2555:
7224:Stochastic chains with memory of variable length
6774:, (1984) Macmillan Publishing Company, New York
6091:if the bits are not equal, output the first bit.
4012:{\displaystyle f:{\mathcal {B}}\to \mathbb {R} }
3719:{\displaystyle f:{\mathcal {B}}\to \mathbb {R} }
3681:{\displaystyle P:{\mathcal {B}}\to \mathbb {R} }
2011:
6384:Example: The input stream from the AMLS paper,
3974:{\displaystyle {\mathcal {B}}\to \mathbb {R} .}
3888:{\displaystyle {\mathcal {B}}\to \mathbb {R} .}
2243:states that the average of the sequence, i.e.,
6536:()()(1)()(1)()(1)(1)()()(0)(0)()(0)(1)(1)()(1)
6076:, which actually extracts uniform randomness.
2434:{\displaystyle \mathbb {E} =\mathbb {P} ()=p,}
2335:, assumed to be represented by 1, is given by
6813:
3625:{\displaystyle P(T^{-1}(\sigma ))=P(\sigma )}
434:) is a finite or infinite sequence of binary
399:
8:
5928:
5886:
5512:
5499:
3036:, and this is the simplest example thereof.
1206:
1187:
1153:
1134:
1104:
1086:
1011:. This algebra is then commonly written as
915:
902:
861:
849:
806:
794:
6142:For example, an input stream of eight bits
4890:; for the doubly-infinite sequence of bits
3688:, consider instead some arbitrary function
2045:{\displaystyle \lim _{n\to \infty }p^{n}=0}
7352:Autoregressiveâmoving-average (ARMA) model
6820:
6806:
6798:
2938:
2816:{\displaystyle S_{n}=\sum _{i=1}^{n}X_{i}}
2130:{\displaystyle (\Omega ,{\mathcal {B}},P)}
1220:{\displaystyle P=\{p,1-p\}^{\mathbb {Z} }}
1167:{\displaystyle P=\{p,1-p\}^{\mathbb {N} }}
406:
392:
93:
6685:
6276:Learn how and when to remove this message
6030:
6026:
6025:
6022:
5997:
5996:
5994:
5973:
5969:
5968:
5965:
5931:
5907:
5896:
5895:
5877:
5873:
5872:
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5788:
5777:
5733:
5714:
5655:
5636:
5598:
5575:
5517:
5516:
5515:
5497:
5461:
5451:
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5439:
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5416:
5375:
5362:
5333:
5323:
5309:
5292:
5282:
5268:
5266:
5238:
5232:
5208:
5195:
5182:
5172:
5166:
5165:
5162:
5134:
5128:
5127:
5124:
5084:
5067:
5050:
5033:
5007:
5001:
5000:
4992:
4960:
4940:
4921:{\displaystyle \Omega =2^{\mathbb {Z} },}
4909:
4908:
4907:
4895:
4867:
4863:
4840:
4807:
4794:
4769:
4756:
4743:
4731:
4711:
4687:
4655:
4635:
4600:
4590:
4584:
4578:
4567:
4555:
4525:
4512:
4506:
4477:
4473:
4459:
4418:
4412:
4411:
4408:
4367:
4361:
4360:
4357:
4317:
4291:
4261:
4255:
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4219:
4189:
4183:
4182:
4163:
4157:
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4131:
4125:
4124:
4121:
4096:
4090:
4089:
4086:
4059:
4037:
4031:
4030:
4027:
4005:
4004:
3995:
3994:
3986:
3964:
3963:
3954:
3953:
3951:
3930:
3924:
3923:
3920:
3900:
3878:
3877:
3868:
3867:
3865:
3829:
3793:
3776:
3749:
3737:
3712:
3711:
3702:
3701:
3693:
3674:
3673:
3664:
3663:
3655:
3586:
3574:
3559:{\displaystyle \sigma \in {\mathcal {B}}}
3550:
3549:
3541:
3508:
3495:
3470:
3457:
3444:
3432:
3405:
3404:
3403:
3391:
3256:
3222:
3204:
3175:
3137:
3113:
3107:
3086:
3080:
2984:
2974:
2973:
2953:
2943:
2925:
2914:
2881:
2860:
2854:
2807:
2797:
2786:
2773:
2767:
2734:
2712:
2701:
2688:
2686:
2665:
2651:
2650:
2648:
2577:
2567:
2554:
2552:
2529:
2455:
2449:
2404:
2390:
2389:
2377:
2366:
2365:
2363:
2340:
2307:
2297:
2286:
2272:
2263:
2252:
2251:
2248:
2224:
2204:
2184:
2164:
2159:Let us assume the canonical process with
2112:
2111:
2100:
2057:
2030:
2014:
2008:
1981:
1954:
1948:
1928:
1901:
1895:
1875:
1841:
1825:
1819:
1794:
1788:
1758:
1736:
1720:
1707:
1688:
1675:
1662:
1649:
1637:
1584:
1562:
1543:
1524:
1511:
1496:
1455:
1431:
1415:
1402:
1393:
1317:
1258:
1211:
1210:
1209:
1179:
1158:
1157:
1156:
1126:
1084:
1055:
1054:
1052:
1028:
1027:
1016:
955:
954:
953:
941:
920:
919:
918:
893:
892:
891:
879:
841:
786:
554:
82:Learn how and when to remove this message
6252:Relevant discussion may be found on the
5830:in this case, otherwise, it is merely a
5806:(the "fair coin"); otherwise not. Thus,
4394:{\displaystyle {\mathcal {L}}_{T}(P)=P.}
3414:{\displaystyle \Omega =2^{\mathbb {N} }}
1040:{\displaystyle (\Omega ,{\mathcal {B}})}
964:{\displaystyle \Omega =2^{\mathbb {Z} }}
45:This article includes a list of general
6663:
6661:
6567:
6208:Von Neumann (classical) main operation
6095:This table summarizes the computation.
5846:is often used informally to refer to a
5119:Restricting the action of the operator
4935:Consider now the space of functions in
3016:Inserting this into the expression for
456:identically distributed and independent
104:
7658:Doob's martingale convergence theorems
5526:{\displaystyle \{H,T\}^{\mathbb {N} }}
4650:is a real number in the unit interval
4486:{\displaystyle x\mapsto 2x{\bmod {1}}}
475:or experiment. They all have the same
7410:Constant elasticity of variance (CEV)
7400:ChanâKarolyiâLongstaffâSanders (CKLS)
6216:if (Bit1 â Bit2) { output(Bit1) }
4332:. This linear operator is called the
2902:. In this case, one may make use of
734:The number of failures needed to get
715:The number of successes in the first
471:in the sequence is associated with a
7:
6573:
6571:
2475:that compose the Bernoulli process.
1599:{\displaystyle P()=p^{k}(1-p)^{n-k}}
502:is a finite or infinite sequence of
6181:. More generally, it works for any
5828:measure preserving dynamical system
4879:{\displaystyle T(y)=2y{\bmod {1}}.}
4540:{\displaystyle b_{0},b_{1},\cdots }
3650:Instead of the probability measure
3354:measure-preserving dynamical system
1381:We denote this distribution by Ber(
588:independent identically distributed
7897:Skorokhod's representation theorem
7678:Law of large numbers (weak/strong)
5440:
5143:{\displaystyle {\mathcal {L}}_{T}}
4897:
4579:
4427:{\displaystyle {\mathcal {L}}_{T}}
4105:{\displaystyle {\mathcal {L}}_{T}}
3939:{\displaystyle {\mathcal {L}}_{T}}
3393:
3183:
2889:
2693:
2559:
2105:
2021:
1021:
943:
881:
51:it lacks sufficient corresponding
25:
7867:Martingale representation theorem
5960:So defined, a Bernoulli sequence
5570:In this case, the transformation
3041:asymptotic equipartition property
2090:asymptotic equipartition property
7912:Stochastic differential equation
7802:Doob's optional stopping theorem
7797:DoobâMeyer decomposition theorem
6756:from the original on 2019-02-12.
6229:
6039:{\displaystyle \mathbb {Z} ^{x}}
5982:{\displaystyle \mathbb {Z} ^{x}}
4928:the induced homomorphism is the
4338:RuelleâFrobeniusâPerron operator
3321:isomorphism of dynamical systems
3319:to another, in the sense of the
2471:out of the infinite sequence of
1370:{\displaystyle pX(0)=P(X=0)=1-p}
1117:, then one can define a natural
623:The two possible values of each
440:discrete-time stochastic process
112:
36:
7782:Convergence of random variables
7668:FisherâTippettâGnedenko theorem
6714:from the original on 2010-03-31
6258:check for citation inaccuracies
6146:would by grouped into pairs as
6088:if the bits are equal, discard;
2510:) of such strings that contain
1121:on the product space, given by
707:. Several random variables and
488:checking whether a coin is fair
7380:Binomial options pricing model
6221:Iterated von Neumann extractor
5919:
5913:
5387:
5381:
5027:
5021:
4971:
4965:
4851:
4845:
4819:
4787:
4781:
4736:
4726:, on the unit interval. Since
4675:{\displaystyle 0\leq y\leq 1.}
4464:
4379:
4373:
4273:
4267:
4244:
4235:
4201:
4195:
4175:
4169:
4149:
4137:
4001:
3960:
3946:on the space of all functions
3874:
3847:
3844:
3838:
3822:
3813:
3807:
3708:
3670:
3619:
3613:
3604:
3601:
3595:
3579:
3520:
3488:
3482:
3437:
3297:of a Bernoulli process. Here,
3277:
3265:
3249:
3237:
3180:
2886:
2731:
2718:
2680:
2677:
2658:
2655:
2606:
2594:
2546:
2534:
2444:for any given random variable
2419:
2416:
2397:
2394:
2383:
2370:
2257:
2124:
2102:
2018:
1853:
1834:
1755:
1742:
1726:
1642:
1581:
1568:
1552:
1549:
1504:
1501:
1437:
1395:
1352:
1340:
1331:
1325:
1305:{\displaystyle pX(1)=P(X=1)=p}
1293:
1281:
1272:
1266:
1064:{\displaystyle {\mathcal {B}}}
1034:
1018:
740:negative binomial distribution
179:Collectively exhaustive events
1:
7847:Kolmogorov continuity theorem
7683:Law of the iterated logarithm
3761:{\displaystyle f\circ T^{-1}}
3032:; this is the content of the
2842:is exactly a special case of
1969:{\displaystyle \omega _{i}=T}
1916:{\displaystyle \omega _{i}=H}
1617:appears in the sequence, and
1481:{\displaystyle 1,2,\cdots ,n}
600:. Given that the probability
7852:Kolmogorov extension theorem
7531:Generalized queueing network
7039:Interacting particle systems
6414:(11)(00)(10)(11)(10)(11)(10)
6193: + (1 â
6004:{\displaystyle \mathbb {N} }
3364:. These are reviewed below.
3325:Ornstein isomorphism theorem
3189:{\displaystyle n\to \infty }
2906:to the factorial, and write
2895:{\displaystyle n\to \infty }
2077:{\displaystyle 0\leq p<1}
1625:is the number of times that
1613:is the number of times that
6984:Continuous-time random walk
6668:Peres, Yuval (March 1992).
6080:Basic von Neumann extractor
4350:FrobeniusâPerron eigenvalue
4344:, that is, a collection of
582: = 1 is the same.
8107:
7992:Extreme value theory (EVT)
7792:Doob decomposition theorem
7084:OrnsteinâUhlenbeck process
6855:Chinese restaurant process
6639:(letter) L483-L485 (1992).
6057:
5857:of coin flips, there is a
5544:
3371:
3075:. The size of this set is
2838:. So we can know that the
2144:
975:on this space, called the
815:{\displaystyle 2=\{H,T\}.}
8060:
7872:Optional stopping theorem
7673:Large deviation principle
7425:HeathâJarrowâMorton (HJM)
7362:Moving-average (MA) model
7347:Autoregressive (AR) model
7172:Hidden Markov model (HMM)
7106:SchrammâLoewner evolution
4116:, as (obviously) one has
3360:, and the other is as an
1870:notation, meaning either
1248:probability mass function
1110:{\displaystyle \{p,1-p\}}
867:{\displaystyle 2=\{H,T\}}
709:probability distributions
7787:Doléans-Dade exponential
7617:Progressively measurable
7415:CoxâIngersollâRoss (CIR)
6747:10.1109/HST.2016.7495553
6674:The Annals of Statistics
6551:
6214:
6068: = 1/2 by the
5533:is sometimes called the
2904:Stirling's approximation
1229:discrete random variable
349:Law of total probability
344:Conditional independence
233:Exponential distribution
218:Probability distribution
8007:Mathematical statistics
7997:Large deviations theory
7827:Infinitesimal generator
7688:Maximal ergodic theorem
7607:Piecewise-deterministic
7209:Random dynamical system
7074:Markov additive process
6702:"Tossing a Biased Coin"
4886:This map is called the
4454: : [0,1) â [0,1),
4340:. This operator has a
3860:is again some function
3151:{\displaystyle H\leq 1}
1996:is commonly called the
738:successes, which has a
328:Conditional probability
66:more precise citations.
7842:KarhunenâLoĂšve theorem
7777:CameronâMartin formula
7741:BurkholderâDavisâGundy
7136:Variance gamma process
6687:10.1214/aos/1176348543
6603:Klenke, Achim (2006).
6431:(10)(11)(11)(01)(01)()
6040:
6005:
5983:
5936:
5820:
5800:
5763:
5584:
5527:
5479:
5444:
5394:
5248:
5218:
5144:
5110:
4978:
4949:
4922:
4880:
4829:
4720:
4696:
4676:
4644:
4621:
4583:
4541:
4498:
4487:
4428:
4395:
4326:
4306:
4280:
4208:
4106:
4072:
4013:
3975:
3940:
3909:
3889:
3854:
3762:
3720:
3682:
3647:on the product space.
3626:
3560:
3527:
3415:
3348:, as an example of an
3284:
3190:
3152:
3126:
3125:{\displaystyle 2^{nH}}
3096:
3007:
2896:
2870:
2840:Bernoulli distribution
2836:Bernoulli distribution
2817:
2802:
2753:
2619:
2465:
2435:
2349:
2317:
2302:
2233:
2213:
2193:
2173:
2131:
2078:
2046:
1990:
1970:
1937:
1917:
1884:
1860:
1859:{\displaystyle x_{i}=}
1804:
1774:
1600:
1482:
1444:
1371:
1306:
1236:Bernoulli distribution
1221:
1168:
1111:
1065:
1047:where the elements of
1041:
965:
930:
868:
816:
755:geometric distribution
705:Bernoulli distribution
563:
546:is either 0 or 1;
529:, ..., such that
477:Bernoulli distribution
270:Continuous or discrete
223:Bernoulli distribution
7972:Actuarial mathematics
7934:Uniform integrability
7929:Stratonovich integral
7857:LĂ©vyâProkhorov metric
7761:MarcinkiewiczâZygmund
7648:Central limit theorem
7250:Gaussian random field
7079:McKeanâVlasov process
6999:Dyson Brownian motion
6860:GaltonâWatson process
6420:(1)(1)(0)(1)(0)(1)(0)
6183:exchangeable sequence
6070:von Neumann extractor
6054:Randomness extraction
6041:
6006:
5984:
5937:
5821:
5801:
5799:{\displaystyle p=1/2}
5764:
5585:
5528:
5480:
5424:
5395:
5256:Bernoulli polynomials
5249:
5247:{\displaystyle B_{n}}
5219:
5145:
5111:
4979:
4950:
4923:
4888:dyadic transformation
4881:
4830:
4721:
4697:
4677:
4645:
4622:
4563:
4542:
4488:
4449:
4436:Bernoulli polynomials
4429:
4396:
4327:
4307:
4281:
4209:
4107:
4073:
4014:
3976:
3941:
3910:
3890:
3855:
3763:
3721:
3683:
3627:
3561:
3528:
3416:
3378:Dyadic transformation
3285:
3191:
3153:
3127:
3097:
3095:{\displaystyle 2^{n}}
3034:central limit theorem
3008:
2897:
2871:
2869:{\displaystyle S_{n}}
2844:Binomial distribution
2832:Binomial distribution
2825:Binomial distribution
2818:
2782:
2754:
2620:
2466:
2464:{\displaystyle X_{i}}
2436:
2350:
2318:
2282:
2234:
2214:
2194:
2174:
2155:Binomial distribution
2151:Central limit theorem
2132:
2079:
2047:
1991:
1971:
1938:
1918:
1885:
1861:
1805:
1803:{\displaystyle X_{i}}
1775:
1601:
1483:
1445:
1372:
1307:
1222:
1169:
1112:
1066:
1042:
995:stands for heads and
971:. There is a natural
966:
936:or the two-sided set
931:
869:
817:
721:binomial distribution
564:
228:Binomial distribution
8091:Stochastic processes
8047:Time series analysis
8002:Mathematical finance
7887:Reflection principle
7214:Regenerative process
7014:FlemingâViot process
6829:Stochastic processes
6707:. eecs.harvard.edu.
6633:Journal of Physics A
6074:randomness extractor
6060:Randomness extractor
6021:
6017:Bernoulli sequences
5993:
5964:
5868:
5810:
5776:
5597:
5574:
5496:
5415:
5265:
5231:
5161:
5123:
4991:
4977:{\displaystyle f(y)}
4959:
4939:
4894:
4839:
4730:
4710:
4686:
4654:
4634:
4554:
4505:
4458:
4407:
4356:
4316:
4290:
4218:
4120:
4085:
4026:
3985:
3981:That is, given some
3950:
3919:
3915:induces another map
3899:
3864:
3775:
3736:
3692:
3654:
3573:
3540:
3431:
3390:
3352:and specifically, a
3203:
3174:
3136:
3106:
3079:
2913:
2880:
2853:
2846:when n equals to 1.
2766:
2647:
2528:
2520:binomial coefficient
2448:
2362:
2339:
2323:, will approach the
2247:
2241:law of large numbers
2223:
2203:
2183:
2163:
2147:Law of large numbers
2137:, as defined above.
2099:
2056:
2007:
1980:
1976:. This probability
1947:
1927:
1894:
1874:
1818:
1787:
1636:
1495:
1454:
1392:
1316:
1257:
1178:
1125:
1083:
1051:
1015:
940:
878:
840:
785:
719:trials, which has a
553:
354:Law of large numbers
323:Marginal probability
248:Poisson distribution
97:Part of a series on
8042:Stochastic analysis
7882:Quadratic variation
7877:Prokhorov's theorem
7812:FeynmanâKac formula
7282:Markov random field
6930:Birthâdeath process
6607:. Springer-Verlag.
6423:(1)(0)()(1)()(1)()
5832:conservative system
5561:group (mathematics)
4305:{\displaystyle f,g}
3030:Normal distribution
3028:), one obtains the
2355:. In fact, one has
1810:is a binary-valued
444:Bernoulli variables
313:Complementary event
255:Probability measure
243:Pareto distribution
238:Normal distribution
8012:Probability theory
7892:Skorokhod integral
7862:Malliavin calculus
7445:Korn-Kreer-Lenssen
7329:Time series models
7292:PitmanâYor process
6770:Carl W. Helstrom,
6605:Probability Theory
6582:. pp. 45â46.
6406:new sequence 2(1)
6311:new sequence 2(1)
6036:
6001:
5979:
5947:Bernoulli sequence
5932:
5844:Bernoulli sequence
5838:Bernoulli sequence
5816:
5796:
5759:
5580:
5523:
5475:
5408:Note that the sum
5390:
5244:
5214:
5140:
5106:
4984:one can find that
4974:
4945:
4918:
4876:
4825:
4716:
4692:
4672:
4640:
4617:
4537:
4499:
4483:
4424:
4391:
4322:
4302:
4276:
4204:
4102:
4068:
4009:
3971:
3936:
3905:
3885:
3850:
3758:
3716:
3678:
3622:
3556:
3523:
3411:
3293:This value is the
3280:
3186:
3148:
3122:
3092:
3062:Kolmogorov 0-1 law
3003:
2892:
2866:
2813:
2749:
2615:
2461:
2431:
2345:
2313:
2229:
2209:
2189:
2169:
2127:
2074:
2042:
2025:
1986:
1966:
1933:
1913:
1880:
1856:
1800:
1770:
1596:
1478:
1440:
1367:
1302:
1217:
1164:
1107:
1061:
1037:
1007:, specifically, a
961:
926:
864:
831:countably infinite
812:
779:probability spaces
652:with parameter p.
569:, the probability
559:
549:for all values of
364:Boole's inequality
300:Stochastic process
189:Mutual exclusivity
106:Probability theory
18:Bernoulli sequence
8078:
8077:
8032:Signal processing
7751:Doob's upcrossing
7746:Doob's martingale
7710:EngelbertâSchmidt
7653:Donsker's theorem
7587:Feller-continuous
7455:RendlemanâBartter
7245:Dirichlet process
7162:Branching process
7131:Telegraph process
7024:Geometric process
7004:Empirical process
6994:Diffusion process
6850:Branching process
6845:Bernoulli process
6627:Pierre Gaspard, "
6614:978-1-84800-047-6
6529:
6528:
6417:()()(1)()(1)()(1)
6403:new sequence 1(A)
6379:
6378:
6308:new sequence 1(A)
6286:
6285:
6278:
6169:) = (1â
6140:
6139:
6048:ergodic sequences
5819:{\displaystyle T}
5583:{\displaystyle T}
5565:topological group
5473:
5349:
5317:
5300:
5276:
5152:discrete spectrum
5100:
5075:
5058:
5041:
4948:{\displaystyle y}
4835:one can see that
4719:{\displaystyle T}
4695:{\displaystyle T}
4643:{\displaystyle y}
4612:
4403:If one restricts
4334:transfer operator
4325:{\displaystyle a}
3908:{\displaystyle T}
3645:invariant measure
3637:Bernoulli measure
3340:Dynamical systems
3295:Bernoulli entropy
2992:
2936:
2834:will turn into a
2700:
2613:
2566:
2482:in a sequence of
2348:{\displaystyle p}
2329:expectation value
2280:
2260:
2232:{\displaystyle 0}
2212:{\displaystyle T}
2192:{\displaystyle 1}
2172:{\displaystyle H}
2010:
1998:Bernoulli measure
1989:{\displaystyle P}
1936:{\displaystyle 0}
1883:{\displaystyle 1}
1075:Bernoulli measure
773:Formal definition
500:Bernoulli process
428:Bernoulli process
416:
415:
318:Joint probability
265:Bernoulli process
164:Probability space
92:
91:
84:
16:(Redirected from
8098:
8052:Machine learning
7939:Usual hypotheses
7822:Girsanov theorem
7807:Dynkin's formula
7572:Continuous paths
7480:Actuarial models
7420:GarmanâKohlhagen
7390:BlackâKarasinski
7385:BlackâDermanâToy
7372:Financial models
7238:Fields and other
7167:Gaussian process
7116:Sigma-martingale
6920:Additive process
6822:
6815:
6808:
6799:
6758:
6757:
6755:
6740:
6729:
6723:
6722:
6720:
6719:
6713:
6706:
6698:
6692:
6691:
6689:
6665:
6656:
6646:
6640:
6625:
6619:
6618:
6600:
6594:
6593:
6575:
6391:
6390:
6299:
6298:
6281:
6274:
6270:
6267:
6261:
6233:
6232:
6225:
6148:(10)(01)(10)(11)
6098:
6097:
6045:
6043:
6042:
6037:
6035:
6034:
6029:
6010:
6008:
6007:
6002:
6000:
5988:
5986:
5985:
5980:
5978:
5977:
5972:
5941:
5939:
5938:
5933:
5912:
5911:
5899:
5882:
5881:
5876:
5825:
5823:
5822:
5817:
5805:
5803:
5802:
5797:
5792:
5768:
5766:
5765:
5760:
5755:
5751:
5744:
5743:
5725:
5724:
5677:
5673:
5666:
5665:
5647:
5646:
5589:
5587:
5586:
5581:
5532:
5530:
5529:
5524:
5522:
5521:
5520:
5484:
5482:
5481:
5476:
5474:
5472:
5471:
5456:
5455:
5446:
5443:
5438:
5399:
5397:
5396:
5391:
5380:
5379:
5370:
5369:
5354:
5350:
5345:
5334:
5328:
5327:
5318:
5310:
5305:
5301:
5293:
5287:
5286:
5277:
5269:
5253:
5251:
5250:
5245:
5243:
5242:
5223:
5221:
5220:
5215:
5213:
5212:
5203:
5202:
5187:
5186:
5177:
5176:
5171:
5170:
5149:
5147:
5146:
5141:
5139:
5138:
5133:
5132:
5115:
5113:
5112:
5107:
5105:
5101:
5096:
5085:
5076:
5068:
5063:
5059:
5051:
5042:
5034:
5020:
5016:
5012:
5011:
5006:
5005:
4983:
4981:
4980:
4975:
4954:
4952:
4951:
4946:
4927:
4925:
4924:
4919:
4914:
4913:
4912:
4885:
4883:
4882:
4877:
4872:
4871:
4834:
4832:
4831:
4826:
4812:
4811:
4799:
4798:
4774:
4773:
4761:
4760:
4748:
4747:
4725:
4723:
4722:
4717:
4701:
4699:
4698:
4693:
4681:
4679:
4678:
4673:
4649:
4647:
4646:
4641:
4626:
4624:
4623:
4618:
4613:
4611:
4610:
4595:
4594:
4585:
4582:
4577:
4546:
4544:
4543:
4538:
4530:
4529:
4517:
4516:
4495:Lebesgue measure
4492:
4490:
4489:
4484:
4482:
4481:
4442:The 2x mod 1 map
4433:
4431:
4430:
4425:
4423:
4422:
4417:
4416:
4400:
4398:
4397:
4392:
4372:
4371:
4366:
4365:
4331:
4329:
4328:
4323:
4311:
4309:
4308:
4303:
4285:
4283:
4282:
4277:
4266:
4265:
4260:
4259:
4234:
4233:
4228:
4227:
4213:
4211:
4210:
4205:
4194:
4193:
4188:
4187:
4168:
4167:
4162:
4161:
4136:
4135:
4130:
4129:
4111:
4109:
4108:
4103:
4101:
4100:
4095:
4094:
4077:
4075:
4074:
4069:
4067:
4066:
4042:
4041:
4036:
4035:
4018:
4016:
4015:
4010:
4008:
4000:
3999:
3980:
3978:
3977:
3972:
3967:
3959:
3958:
3945:
3943:
3942:
3937:
3935:
3934:
3929:
3928:
3914:
3912:
3911:
3906:
3894:
3892:
3891:
3886:
3881:
3873:
3872:
3859:
3857:
3856:
3851:
3837:
3836:
3806:
3802:
3801:
3800:
3767:
3765:
3764:
3759:
3757:
3756:
3725:
3723:
3722:
3717:
3715:
3707:
3706:
3687:
3685:
3684:
3679:
3677:
3669:
3668:
3631:
3629:
3628:
3623:
3594:
3593:
3565:
3563:
3562:
3557:
3555:
3554:
3532:
3530:
3529:
3524:
3513:
3512:
3500:
3499:
3475:
3474:
3462:
3461:
3449:
3448:
3420:
3418:
3417:
3412:
3410:
3409:
3408:
3374:Bernoulli scheme
3346:dynamical system
3313:John von Neumann
3289:
3287:
3286:
3281:
3261:
3260:
3227:
3226:
3195:
3193:
3192:
3187:
3157:
3155:
3154:
3149:
3131:
3129:
3128:
3123:
3121:
3120:
3101:
3099:
3098:
3093:
3091:
3090:
3012:
3010:
3009:
3004:
3002:
2998:
2997:
2993:
2985:
2979:
2978:
2961:
2960:
2948:
2947:
2937:
2926:
2901:
2899:
2898:
2893:
2875:
2873:
2872:
2867:
2865:
2864:
2822:
2820:
2819:
2814:
2812:
2811:
2801:
2796:
2778:
2777:
2758:
2756:
2755:
2750:
2745:
2744:
2717:
2716:
2707:
2706:
2705:
2692:
2670:
2669:
2654:
2624:
2622:
2621:
2616:
2614:
2612:
2586:
2578:
2573:
2572:
2571:
2558:
2518:is given by the
2473:Bernoulli trials
2470:
2468:
2467:
2462:
2460:
2459:
2440:
2438:
2437:
2432:
2409:
2408:
2393:
2382:
2381:
2369:
2354:
2352:
2351:
2346:
2322:
2320:
2319:
2314:
2312:
2311:
2301:
2296:
2281:
2273:
2268:
2267:
2262:
2261:
2253:
2238:
2236:
2235:
2230:
2218:
2216:
2215:
2210:
2198:
2196:
2195:
2190:
2178:
2176:
2175:
2170:
2136:
2134:
2133:
2128:
2117:
2116:
2083:
2081:
2080:
2075:
2051:
2049:
2048:
2043:
2035:
2034:
2024:
1995:
1993:
1992:
1987:
1975:
1973:
1972:
1967:
1959:
1958:
1942:
1940:
1939:
1934:
1922:
1920:
1919:
1914:
1906:
1905:
1889:
1887:
1886:
1881:
1865:
1863:
1862:
1857:
1846:
1845:
1830:
1829:
1809:
1807:
1806:
1801:
1799:
1798:
1779:
1777:
1776:
1771:
1769:
1768:
1741:
1740:
1725:
1724:
1712:
1711:
1693:
1692:
1680:
1679:
1667:
1666:
1654:
1653:
1605:
1603:
1602:
1597:
1595:
1594:
1567:
1566:
1548:
1547:
1529:
1528:
1516:
1515:
1487:
1485:
1484:
1479:
1449:
1447:
1446:
1443:{\displaystyle }
1441:
1436:
1435:
1420:
1419:
1407:
1406:
1376:
1374:
1373:
1368:
1311:
1309:
1308:
1303:
1226:
1224:
1223:
1218:
1216:
1215:
1214:
1173:
1171:
1170:
1165:
1163:
1162:
1161:
1116:
1114:
1113:
1108:
1070:
1068:
1067:
1062:
1060:
1059:
1046:
1044:
1043:
1038:
1033:
1032:
977:product topology
970:
968:
967:
962:
960:
959:
958:
935:
933:
932:
927:
925:
924:
923:
898:
897:
896:
873:
871:
870:
865:
821:
819:
818:
813:
669:, ...
650:Bernoulli trials
591:Bernoulli trials
568:
566:
565:
560:
507:random variables
481:Bernoulli scheme
436:random variables
408:
401:
394:
184:Elementary event
116:
94:
87:
80:
76:
73:
67:
62:this article by
53:inline citations
40:
39:
32:
21:
8106:
8105:
8101:
8100:
8099:
8097:
8096:
8095:
8081:
8080:
8079:
8074:
8056:
8017:Queueing theory
7960:
7902:Skorokhod space
7765:
7756:KunitaâWatanabe
7727:
7693:Sanov's theorem
7663:Ergodic theorem
7636:
7632:Time-reversible
7550:
7513:Queueing models
7507:
7503:SparreâAnderson
7493:CramĂ©râLundberg
7474:
7460:SABR volatility
7366:
7323:
7275:Boolean network
7233:
7219:Renewal process
7150:
7099:Non-homogeneous
7089:Poisson process
6979:Contact process
6942:Brownian motion
6912:Continuous time
6906:
6900:Maximal entropy
6831:
6826:
6789:
6767:
6765:Further reading
6762:
6761:
6753:
6738:
6731:
6730:
6726:
6717:
6715:
6711:
6704:
6700:
6699:
6695:
6667:
6666:
6659:
6647:
6643:
6626:
6622:
6615:
6602:
6601:
6597:
6590:
6577:
6576:
6569:
6564:
6555:
6554:
6531:
6437:(0)(1)(1)(0)(0)
6282:
6271:
6265:
6262:
6251:
6234:
6230:
6223:
6218:
6217:
6082:
6072:, the earliest
6062:
6056:
6024:
6019:
6018:
5991:
5990:
5967:
5962:
5961:
5903:
5871:
5866:
5865:
5840:
5808:
5807:
5774:
5773:
5729:
5710:
5685:
5681:
5651:
5632:
5607:
5603:
5595:
5594:
5572:
5571:
5549:
5547:Markov odometer
5543:
5511:
5494:
5493:
5490:Cantor function
5457:
5447:
5413:
5412:
5406:
5371:
5358:
5335:
5329:
5319:
5288:
5278:
5263:
5262:
5234:
5229:
5228:
5204:
5191:
5178:
5164:
5159:
5158:
5126:
5121:
5120:
5086:
5080:
5046:
4999:
4998:
4994:
4989:
4988:
4957:
4956:
4937:
4936:
4903:
4892:
4891:
4837:
4836:
4803:
4790:
4765:
4752:
4739:
4728:
4727:
4708:
4707:
4684:
4683:
4652:
4651:
4632:
4631:
4596:
4586:
4552:
4551:
4521:
4508:
4503:
4502:
4456:
4455:
4444:
4410:
4405:
4404:
4359:
4354:
4353:
4314:
4313:
4288:
4287:
4253:
4221:
4216:
4215:
4181:
4155:
4123:
4118:
4117:
4114:linear operator
4088:
4083:
4082:
4055:
4029:
4024:
4023:
3983:
3982:
3948:
3947:
3922:
3917:
3916:
3897:
3896:
3862:
3861:
3825:
3789:
3782:
3778:
3773:
3772:
3745:
3734:
3733:
3690:
3689:
3652:
3651:
3582:
3571:
3570:
3538:
3537:
3504:
3491:
3466:
3453:
3440:
3429:
3428:
3399:
3388:
3387:
3380:
3372:Main articles:
3370:
3368:Bernoulli shift
3342:
3252:
3218:
3201:
3200:
3196:one finds that
3172:
3171:
3134:
3133:
3109:
3104:
3103:
3082:
3077:
3076:
2980:
2966:
2962:
2949:
2939:
2911:
2910:
2878:
2877:
2856:
2851:
2850:
2803:
2769:
2764:
2763:
2730:
2708:
2687:
2661:
2645:
2644:
2587:
2579:
2553:
2526:
2525:
2514:occurrences of
2451:
2446:
2445:
2400:
2373:
2360:
2359:
2337:
2336:
2303:
2250:
2245:
2244:
2221:
2220:
2219:represented by
2201:
2200:
2181:
2180:
2179:represented by
2161:
2160:
2157:
2145:Main articles:
2143:
2097:
2096:
2054:
2053:
2026:
2005:
2004:
1978:
1977:
1950:
1945:
1944:
1925:
1924:
1897:
1892:
1891:
1872:
1871:
1868:Iverson bracket
1837:
1821:
1816:
1815:
1812:random variable
1790:
1785:
1784:
1754:
1732:
1716:
1703:
1684:
1671:
1658:
1645:
1634:
1633:
1580:
1558:
1539:
1520:
1507:
1493:
1492:
1452:
1451:
1427:
1411:
1398:
1390:
1389:
1314:
1313:
1255:
1254:
1238:with parameter
1205:
1176:
1175:
1152:
1123:
1122:
1081:
1080:
1077:
1049:
1048:
1013:
1012:
949:
938:
937:
914:
887:
876:
875:
838:
837:
827:
783:
782:
775:
695:
686:
675:
668:
661:
647:
631:
621:
581:
551:
550:
545:
537:, the value of
528:
521:
514:
496:
473:Bernoulli trial
470:
453:
432:Jacob Bernoulli
412:
260:Random variable
211:Bernoulli trial
88:
77:
71:
68:
58:Please help to
57:
41:
37:
30:
23:
22:
15:
12:
11:
5:
8104:
8102:
8094:
8093:
8083:
8082:
8076:
8075:
8073:
8072:
8067:
8065:List of topics
8061:
8058:
8057:
8055:
8054:
8049:
8044:
8039:
8034:
8029:
8024:
8022:Renewal theory
8019:
8014:
8009:
8004:
7999:
7994:
7989:
7987:Ergodic theory
7984:
7979:
7977:Control theory
7974:
7968:
7966:
7962:
7961:
7959:
7958:
7957:
7956:
7951:
7941:
7936:
7931:
7926:
7921:
7920:
7919:
7909:
7907:Snell envelope
7904:
7899:
7894:
7889:
7884:
7879:
7874:
7869:
7864:
7859:
7854:
7849:
7844:
7839:
7834:
7829:
7824:
7819:
7814:
7809:
7804:
7799:
7794:
7789:
7784:
7779:
7773:
7771:
7767:
7766:
7764:
7763:
7758:
7753:
7748:
7743:
7737:
7735:
7729:
7728:
7726:
7725:
7706:BorelâCantelli
7695:
7690:
7685:
7680:
7675:
7670:
7665:
7660:
7655:
7650:
7644:
7642:
7641:Limit theorems
7638:
7637:
7635:
7634:
7629:
7624:
7619:
7614:
7609:
7604:
7599:
7594:
7589:
7584:
7579:
7574:
7569:
7564:
7558:
7556:
7552:
7551:
7549:
7548:
7543:
7538:
7533:
7528:
7523:
7517:
7515:
7509:
7508:
7506:
7505:
7500:
7495:
7490:
7484:
7482:
7476:
7475:
7473:
7472:
7467:
7462:
7457:
7452:
7447:
7442:
7437:
7432:
7427:
7422:
7417:
7412:
7407:
7402:
7397:
7392:
7387:
7382:
7376:
7374:
7368:
7367:
7365:
7364:
7359:
7354:
7349:
7344:
7339:
7333:
7331:
7325:
7324:
7322:
7321:
7316:
7311:
7310:
7309:
7304:
7294:
7289:
7284:
7279:
7278:
7277:
7272:
7262:
7260:Hopfield model
7257:
7252:
7247:
7241:
7239:
7235:
7234:
7232:
7231:
7226:
7221:
7216:
7211:
7206:
7205:
7204:
7199:
7194:
7189:
7179:
7177:Markov process
7174:
7169:
7164:
7158:
7156:
7152:
7151:
7149:
7148:
7146:Wiener sausage
7143:
7141:Wiener process
7138:
7133:
7128:
7123:
7121:Stable process
7118:
7113:
7111:Semimartingale
7108:
7103:
7102:
7101:
7096:
7086:
7081:
7076:
7071:
7066:
7061:
7056:
7054:Jump diffusion
7051:
7046:
7041:
7036:
7031:
7029:Hawkes process
7026:
7021:
7016:
7011:
7009:Feller process
7006:
7001:
6996:
6991:
6986:
6981:
6976:
6974:Cauchy process
6971:
6970:
6969:
6964:
6959:
6954:
6949:
6939:
6938:
6937:
6927:
6925:Bessel process
6922:
6916:
6914:
6908:
6907:
6905:
6904:
6903:
6902:
6897:
6892:
6887:
6877:
6872:
6867:
6862:
6857:
6852:
6847:
6841:
6839:
6833:
6832:
6827:
6825:
6824:
6817:
6810:
6802:
6796:
6795:
6788:
6787:External links
6785:
6784:
6783:
6766:
6763:
6760:
6759:
6724:
6693:
6680:(1): 590â597.
6657:
6641:
6620:
6613:
6595:
6588:
6566:
6565:
6563:
6560:
6552:
6544:timing attacks
6527:
6526:
6523:
6520:
6517:
6514:
6510:
6509:
6506:
6503:
6500:
6497:
6493:
6492:
6489:
6486:
6483:
6480:
6476:
6475:
6472:
6469:
6466:
6463:
6459:
6458:
6455:
6452:
6449:
6448:(11)(01)(10)()
6446:
6442:
6441:
6438:
6435:
6432:
6429:
6425:
6424:
6421:
6418:
6415:
6412:
6408:
6407:
6404:
6401:
6398:
6395:
6386:11001011101110
6377:
6376:
6373:
6370:
6365:
6361:
6360:
6355:
6352:
6349:
6345:
6344:
6339:
6336:
6333:
6329:
6328:
6325:
6322:
6317:
6313:
6312:
6309:
6306:
6303:
6284:
6283:
6237:
6235:
6228:
6222:
6219:
6215:
6138:
6137:
6134:
6130:
6129:
6126:
6122:
6121:
6118:
6114:
6113:
6110:
6106:
6105:
6102:
6093:
6092:
6089:
6081:
6078:
6058:Main article:
6055:
6052:
6033:
6028:
5999:
5976:
5971:
5943:
5942:
5930:
5927:
5924:
5921:
5918:
5915:
5910:
5906:
5902:
5898:
5894:
5891:
5888:
5885:
5880:
5875:
5839:
5836:
5815:
5795:
5791:
5787:
5784:
5781:
5770:
5769:
5758:
5754:
5750:
5747:
5742:
5739:
5736:
5732:
5728:
5723:
5720:
5717:
5713:
5709:
5706:
5703:
5700:
5697:
5694:
5691:
5688:
5684:
5680:
5676:
5672:
5669:
5664:
5661:
5658:
5654:
5650:
5645:
5642:
5639:
5635:
5631:
5628:
5625:
5622:
5619:
5616:
5613:
5610:
5606:
5602:
5579:
5545:Main article:
5542:
5539:
5519:
5514:
5510:
5507:
5504:
5501:
5486:
5485:
5470:
5467:
5464:
5460:
5454:
5450:
5442:
5437:
5434:
5431:
5427:
5423:
5420:
5405:
5404:The Cantor set
5402:
5401:
5400:
5389:
5386:
5383:
5378:
5374:
5368:
5365:
5361:
5357:
5353:
5348:
5344:
5341:
5338:
5332:
5326:
5322:
5316:
5313:
5308:
5304:
5299:
5296:
5291:
5285:
5281:
5275:
5272:
5241:
5237:
5225:
5224:
5211:
5207:
5201:
5198:
5194:
5190:
5185:
5181:
5175:
5169:
5137:
5131:
5117:
5116:
5104:
5099:
5095:
5092:
5089:
5083:
5079:
5074:
5071:
5066:
5062:
5057:
5054:
5049:
5045:
5040:
5037:
5032:
5029:
5026:
5023:
5019:
5015:
5010:
5004:
4997:
4973:
4970:
4967:
4964:
4944:
4917:
4911:
4906:
4902:
4899:
4875:
4870:
4866:
4862:
4859:
4856:
4853:
4850:
4847:
4844:
4824:
4821:
4818:
4815:
4810:
4806:
4802:
4797:
4793:
4789:
4786:
4783:
4780:
4777:
4772:
4768:
4764:
4759:
4755:
4751:
4746:
4742:
4738:
4735:
4715:
4706:, also called
4691:
4671:
4668:
4665:
4662:
4659:
4639:
4630:The resulting
4628:
4627:
4616:
4609:
4606:
4603:
4599:
4593:
4589:
4581:
4576:
4573:
4570:
4566:
4562:
4559:
4536:
4533:
4528:
4524:
4520:
4515:
4511:
4493:preserves the
4480:
4476:
4472:
4469:
4466:
4463:
4443:
4440:
4421:
4415:
4390:
4387:
4384:
4381:
4378:
4375:
4370:
4364:
4346:eigenfunctions
4321:
4301:
4298:
4295:
4286:for functions
4275:
4272:
4269:
4264:
4258:
4252:
4249:
4246:
4243:
4240:
4237:
4232:
4226:
4203:
4200:
4197:
4192:
4186:
4180:
4177:
4174:
4171:
4166:
4160:
4154:
4151:
4148:
4145:
4142:
4139:
4134:
4128:
4099:
4093:
4079:
4078:
4065:
4062:
4058:
4054:
4051:
4048:
4045:
4040:
4034:
4019:, one defines
4007:
4003:
3998:
3993:
3990:
3970:
3966:
3962:
3957:
3933:
3927:
3904:
3895:Thus, the map
3884:
3880:
3876:
3871:
3849:
3846:
3843:
3840:
3835:
3832:
3828:
3824:
3821:
3818:
3815:
3812:
3809:
3805:
3799:
3796:
3792:
3788:
3785:
3781:
3769:
3768:
3755:
3752:
3748:
3744:
3741:
3714:
3710:
3705:
3700:
3697:
3676:
3672:
3667:
3662:
3659:
3633:
3632:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3597:
3592:
3589:
3585:
3581:
3578:
3553:
3548:
3545:
3534:
3533:
3522:
3519:
3516:
3511:
3507:
3503:
3498:
3494:
3490:
3487:
3484:
3481:
3478:
3473:
3469:
3465:
3460:
3456:
3452:
3447:
3443:
3439:
3436:
3423:shift operator
3407:
3402:
3398:
3395:
3369:
3366:
3350:ergodic system
3341:
3338:
3291:
3290:
3279:
3276:
3273:
3270:
3267:
3264:
3259:
3255:
3251:
3248:
3245:
3242:
3239:
3236:
3233:
3230:
3225:
3221:
3217:
3214:
3211:
3208:
3185:
3182:
3179:
3147:
3144:
3141:
3119:
3116:
3112:
3089:
3085:
3014:
3013:
3001:
2996:
2991:
2988:
2983:
2977:
2972:
2969:
2965:
2959:
2956:
2952:
2946:
2942:
2935:
2932:
2929:
2924:
2921:
2918:
2891:
2888:
2885:
2863:
2859:
2810:
2806:
2800:
2795:
2792:
2789:
2785:
2781:
2776:
2772:
2760:
2759:
2748:
2743:
2740:
2737:
2733:
2729:
2726:
2723:
2720:
2715:
2711:
2704:
2699:
2696:
2691:
2685:
2682:
2679:
2676:
2673:
2668:
2664:
2660:
2657:
2653:
2626:
2625:
2611:
2608:
2605:
2602:
2599:
2596:
2593:
2590:
2585:
2582:
2576:
2570:
2565:
2562:
2557:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2458:
2454:
2442:
2441:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2407:
2403:
2399:
2396:
2392:
2388:
2385:
2380:
2376:
2372:
2368:
2344:
2325:expected value
2310:
2306:
2300:
2295:
2292:
2289:
2285:
2279:
2276:
2271:
2266:
2259:
2256:
2228:
2208:
2188:
2168:
2142:
2139:
2126:
2123:
2120:
2115:
2110:
2107:
2104:
2073:
2070:
2067:
2064:
2061:
2041:
2038:
2033:
2029:
2023:
2020:
2017:
2013:
1985:
1965:
1962:
1957:
1953:
1932:
1912:
1909:
1904:
1900:
1879:
1855:
1852:
1849:
1844:
1840:
1836:
1833:
1828:
1824:
1797:
1793:
1781:
1780:
1767:
1764:
1761:
1757:
1753:
1750:
1747:
1744:
1739:
1735:
1731:
1728:
1723:
1719:
1715:
1710:
1706:
1702:
1699:
1696:
1691:
1687:
1683:
1678:
1674:
1670:
1665:
1661:
1657:
1652:
1648:
1644:
1641:
1607:
1606:
1593:
1590:
1587:
1583:
1579:
1576:
1573:
1570:
1565:
1561:
1557:
1554:
1551:
1546:
1542:
1538:
1535:
1532:
1527:
1523:
1519:
1514:
1510:
1506:
1503:
1500:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1439:
1434:
1430:
1426:
1423:
1418:
1414:
1410:
1405:
1401:
1397:
1379:
1378:
1366:
1363:
1360:
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1321:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1250:is given by
1213:
1208:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1160:
1155:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1076:
1073:
1058:
1036:
1031:
1026:
1023:
1020:
957:
952:
948:
945:
922:
917:
913:
910:
907:
904:
901:
895:
890:
886:
883:
863:
860:
857:
854:
851:
848:
845:
834:direct product
826:
823:
811:
808:
805:
802:
799:
796:
793:
790:
774:
771:
763:
762:
751:
732:
691:
684:
673:
666:
659:
648:may be called
643:
627:
620:
619:Interpretation
617:
584:
583:
577:
562:{\textstyle i}
558:
547:
541:
526:
519:
512:
495:
492:
466:
449:
414:
413:
411:
410:
403:
396:
388:
385:
384:
383:
382:
377:
369:
368:
367:
366:
361:
359:Bayes' theorem
356:
351:
346:
341:
333:
332:
331:
330:
325:
320:
315:
307:
306:
305:
304:
303:
302:
297:
292:
290:Observed value
287:
282:
277:
275:Expected value
272:
267:
257:
252:
251:
250:
245:
240:
235:
230:
225:
215:
214:
213:
203:
202:
201:
196:
191:
186:
181:
171:
166:
158:
157:
156:
155:
150:
145:
144:
143:
133:
132:
131:
118:
117:
109:
108:
102:
101:
90:
89:
72:September 2011
44:
42:
35:
28:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8103:
8092:
8089:
8088:
8086:
8071:
8068:
8066:
8063:
8062:
8059:
8053:
8050:
8048:
8045:
8043:
8040:
8038:
8035:
8033:
8030:
8028:
8025:
8023:
8020:
8018:
8015:
8013:
8010:
8008:
8005:
8003:
8000:
7998:
7995:
7993:
7990:
7988:
7985:
7983:
7980:
7978:
7975:
7973:
7970:
7969:
7967:
7963:
7955:
7952:
7950:
7947:
7946:
7945:
7942:
7940:
7937:
7935:
7932:
7930:
7927:
7925:
7924:Stopping time
7922:
7918:
7915:
7914:
7913:
7910:
7908:
7905:
7903:
7900:
7898:
7895:
7893:
7890:
7888:
7885:
7883:
7880:
7878:
7875:
7873:
7870:
7868:
7865:
7863:
7860:
7858:
7855:
7853:
7850:
7848:
7845:
7843:
7840:
7838:
7835:
7833:
7830:
7828:
7825:
7823:
7820:
7818:
7815:
7813:
7810:
7808:
7805:
7803:
7800:
7798:
7795:
7793:
7790:
7788:
7785:
7783:
7780:
7778:
7775:
7774:
7772:
7768:
7762:
7759:
7757:
7754:
7752:
7749:
7747:
7744:
7742:
7739:
7738:
7736:
7734:
7730:
7723:
7719:
7715:
7714:HewittâSavage
7711:
7707:
7703:
7699:
7698:Zeroâone laws
7696:
7694:
7691:
7689:
7686:
7684:
7681:
7679:
7676:
7674:
7671:
7669:
7666:
7664:
7661:
7659:
7656:
7654:
7651:
7649:
7646:
7645:
7643:
7639:
7633:
7630:
7628:
7625:
7623:
7620:
7618:
7615:
7613:
7610:
7608:
7605:
7603:
7600:
7598:
7595:
7593:
7590:
7588:
7585:
7583:
7580:
7578:
7575:
7573:
7570:
7568:
7565:
7563:
7560:
7559:
7557:
7553:
7547:
7544:
7542:
7539:
7537:
7534:
7532:
7529:
7527:
7524:
7522:
7519:
7518:
7516:
7514:
7510:
7504:
7501:
7499:
7496:
7494:
7491:
7489:
7486:
7485:
7483:
7481:
7477:
7471:
7468:
7466:
7463:
7461:
7458:
7456:
7453:
7451:
7448:
7446:
7443:
7441:
7438:
7436:
7433:
7431:
7428:
7426:
7423:
7421:
7418:
7416:
7413:
7411:
7408:
7406:
7403:
7401:
7398:
7396:
7395:BlackâScholes
7393:
7391:
7388:
7386:
7383:
7381:
7378:
7377:
7375:
7373:
7369:
7363:
7360:
7358:
7355:
7353:
7350:
7348:
7345:
7343:
7340:
7338:
7335:
7334:
7332:
7330:
7326:
7320:
7317:
7315:
7312:
7308:
7305:
7303:
7300:
7299:
7298:
7297:Point process
7295:
7293:
7290:
7288:
7285:
7283:
7280:
7276:
7273:
7271:
7268:
7267:
7266:
7263:
7261:
7258:
7256:
7255:Gibbs measure
7253:
7251:
7248:
7246:
7243:
7242:
7240:
7236:
7230:
7227:
7225:
7222:
7220:
7217:
7215:
7212:
7210:
7207:
7203:
7200:
7198:
7195:
7193:
7190:
7188:
7185:
7184:
7183:
7180:
7178:
7175:
7173:
7170:
7168:
7165:
7163:
7160:
7159:
7157:
7153:
7147:
7144:
7142:
7139:
7137:
7134:
7132:
7129:
7127:
7124:
7122:
7119:
7117:
7114:
7112:
7109:
7107:
7104:
7100:
7097:
7095:
7092:
7091:
7090:
7087:
7085:
7082:
7080:
7077:
7075:
7072:
7070:
7067:
7065:
7062:
7060:
7057:
7055:
7052:
7050:
7047:
7045:
7044:ItĂŽ diffusion
7042:
7040:
7037:
7035:
7032:
7030:
7027:
7025:
7022:
7020:
7019:Gamma process
7017:
7015:
7012:
7010:
7007:
7005:
7002:
7000:
6997:
6995:
6992:
6990:
6987:
6985:
6982:
6980:
6977:
6975:
6972:
6968:
6965:
6963:
6960:
6958:
6955:
6953:
6950:
6948:
6945:
6944:
6943:
6940:
6936:
6933:
6932:
6931:
6928:
6926:
6923:
6921:
6918:
6917:
6915:
6913:
6909:
6901:
6898:
6896:
6893:
6891:
6890:Self-avoiding
6888:
6886:
6883:
6882:
6881:
6878:
6876:
6875:Moran process
6873:
6871:
6868:
6866:
6863:
6861:
6858:
6856:
6853:
6851:
6848:
6846:
6843:
6842:
6840:
6838:
6837:Discrete time
6834:
6830:
6823:
6818:
6816:
6811:
6809:
6804:
6803:
6800:
6794:
6791:
6790:
6786:
6781:
6780:0-02-353560-1
6777:
6773:
6769:
6768:
6764:
6752:
6748:
6744:
6737:
6736:
6728:
6725:
6710:
6703:
6697:
6694:
6688:
6683:
6679:
6675:
6671:
6664:
6662:
6658:
6655:
6654:0-7923-5564-4
6651:
6645:
6642:
6638:
6634:
6630:
6624:
6621:
6616:
6610:
6606:
6599:
6596:
6591:
6589:9781852338961
6585:
6581:
6574:
6572:
6568:
6561:
6559:
6550:
6547:
6545:
6541:
6537:
6532:
6524:
6521:
6518:
6515:
6512:
6511:
6507:
6504:
6501:
6498:
6495:
6494:
6490:
6487:
6484:
6481:
6478:
6477:
6473:
6470:
6467:
6464:
6461:
6460:
6456:
6453:
6450:
6447:
6444:
6443:
6440:()(1)(1)()()
6439:
6436:
6434:(1)()()(0)(0)
6433:
6430:
6427:
6426:
6422:
6419:
6416:
6413:
6410:
6409:
6405:
6402:
6399:
6396:
6393:
6392:
6389:
6387:
6382:
6374:
6371:
6369:
6366:
6363:
6362:
6359:
6356:
6353:
6350:
6347:
6346:
6343:
6340:
6337:
6334:
6331:
6330:
6326:
6323:
6321:
6318:
6315:
6314:
6310:
6307:
6304:
6301:
6300:
6297:
6294:
6290:
6280:
6277:
6269:
6259:
6255:
6249:
6247:
6243:
6238:This section
6236:
6227:
6226:
6220:
6213:
6211:
6206:
6204:
6200:
6196:
6192:
6186:
6184:
6180:
6176:
6172:
6168:
6164:
6159:
6157:
6153:
6149:
6145:
6135:
6132:
6131:
6127:
6124:
6123:
6119:
6116:
6115:
6111:
6108:
6107:
6103:
6100:
6099:
6096:
6090:
6087:
6086:
6085:
6079:
6077:
6075:
6071:
6067:
6061:
6053:
6051:
6049:
6031:
6016:
6012:
5974:
5958:
5956:
5952:
5948:
5925:
5922:
5916:
5908:
5904:
5900:
5892:
5889:
5883:
5878:
5864:
5863:
5862:
5860:
5856:
5851:
5849:
5845:
5837:
5835:
5833:
5829:
5813:
5793:
5789:
5785:
5782:
5779:
5756:
5752:
5748:
5745:
5740:
5737:
5734:
5730:
5726:
5721:
5718:
5715:
5711:
5707:
5704:
5701:
5698:
5695:
5692:
5689:
5686:
5682:
5678:
5674:
5670:
5667:
5662:
5659:
5656:
5652:
5648:
5643:
5640:
5637:
5633:
5629:
5626:
5623:
5620:
5617:
5614:
5611:
5608:
5604:
5600:
5593:
5592:
5591:
5577:
5568:
5566:
5562:
5558:
5554:
5548:
5540:
5538:
5536:
5508:
5505:
5502:
5491:
5468:
5465:
5462:
5458:
5452:
5448:
5435:
5432:
5429:
5425:
5421:
5418:
5411:
5410:
5409:
5403:
5384:
5376:
5372:
5366:
5363:
5359:
5355:
5351:
5346:
5342:
5339:
5336:
5330:
5324:
5320:
5314:
5311:
5306:
5302:
5297:
5294:
5289:
5283:
5279:
5273:
5270:
5261:
5260:
5259:
5257:
5239:
5235:
5209:
5205:
5199:
5196:
5192:
5188:
5183:
5179:
5173:
5157:
5156:
5155:
5153:
5135:
5102:
5097:
5093:
5090:
5087:
5081:
5077:
5072:
5069:
5064:
5060:
5055:
5052:
5047:
5043:
5038:
5035:
5030:
5024:
5017:
5013:
5008:
4995:
4987:
4986:
4985:
4968:
4962:
4955:. Given some
4942:
4933:
4931:
4915:
4904:
4900:
4889:
4873:
4868:
4860:
4857:
4854:
4848:
4842:
4822:
4816:
4813:
4808:
4804:
4800:
4795:
4791:
4784:
4778:
4775:
4770:
4766:
4762:
4757:
4753:
4749:
4744:
4740:
4733:
4713:
4705:
4689:
4669:
4666:
4663:
4660:
4657:
4637:
4614:
4607:
4604:
4601:
4597:
4591:
4587:
4574:
4571:
4568:
4564:
4560:
4557:
4550:
4549:
4548:
4534:
4531:
4526:
4522:
4518:
4513:
4509:
4496:
4478:
4470:
4467:
4461:
4453:
4448:
4441:
4439:
4437:
4419:
4401:
4388:
4385:
4382:
4376:
4368:
4351:
4347:
4343:
4339:
4335:
4319:
4312:and constant
4299:
4296:
4293:
4270:
4262:
4250:
4247:
4241:
4238:
4230:
4198:
4190:
4178:
4172:
4164:
4152:
4146:
4143:
4140:
4132:
4115:
4097:
4063:
4060:
4056:
4052:
4049:
4046:
4043:
4038:
4022:
4021:
4020:
3991:
3988:
3968:
3931:
3902:
3882:
3841:
3833:
3830:
3826:
3819:
3816:
3810:
3803:
3797:
3794:
3790:
3786:
3783:
3779:
3753:
3750:
3746:
3742:
3739:
3732:
3731:
3730:
3729:
3698:
3695:
3660:
3657:
3648:
3646:
3642:
3638:
3635:and thus the
3616:
3610:
3607:
3598:
3590:
3587:
3583:
3576:
3569:
3568:
3567:
3546:
3543:
3517:
3514:
3509:
3505:
3501:
3496:
3492:
3485:
3479:
3476:
3471:
3467:
3463:
3458:
3454:
3450:
3445:
3441:
3434:
3427:
3426:
3425:
3424:
3421:given by the
3400:
3396:
3385:
3379:
3375:
3367:
3365:
3363:
3359:
3355:
3351:
3347:
3339:
3337:
3334:
3330:
3326:
3322:
3318:
3314:
3310:
3308:
3305:standing for
3304:
3300:
3296:
3274:
3271:
3268:
3262:
3257:
3253:
3246:
3243:
3240:
3234:
3231:
3228:
3223:
3219:
3215:
3212:
3209:
3206:
3199:
3198:
3197:
3177:
3169:
3165:
3161:
3145:
3142:
3139:
3117:
3114:
3110:
3087:
3083:
3074:
3070:
3065:
3063:
3059:
3055:
3050:
3046:
3042:
3037:
3035:
3031:
3027:
3023:
3019:
2999:
2994:
2989:
2986:
2981:
2970:
2967:
2963:
2957:
2954:
2950:
2944:
2940:
2933:
2930:
2927:
2922:
2919:
2916:
2909:
2908:
2907:
2905:
2883:
2861:
2857:
2847:
2845:
2841:
2837:
2833:
2828:
2826:
2808:
2804:
2798:
2793:
2790:
2787:
2783:
2779:
2774:
2770:
2746:
2741:
2738:
2735:
2727:
2724:
2721:
2713:
2709:
2697:
2694:
2683:
2674:
2671:
2666:
2662:
2643:
2642:
2641:
2639:
2635:
2631:
2609:
2603:
2600:
2597:
2591:
2588:
2583:
2580:
2574:
2563:
2560:
2549:
2543:
2540:
2537:
2531:
2524:
2523:
2522:
2521:
2517:
2513:
2509:
2505:
2501:
2498:, the number
2497:
2493:
2489:
2485:
2481:
2476:
2474:
2456:
2452:
2428:
2425:
2422:
2413:
2410:
2405:
2401:
2386:
2378:
2374:
2358:
2357:
2356:
2342:
2334:
2330:
2326:
2308:
2304:
2298:
2293:
2290:
2287:
2283:
2277:
2274:
2269:
2264:
2254:
2242:
2226:
2206:
2186:
2166:
2156:
2152:
2148:
2140:
2138:
2121:
2118:
2108:
2093:
2091:
2087:
2071:
2068:
2065:
2062:
2059:
2039:
2036:
2031:
2027:
2015:
2001:
1999:
1983:
1963:
1960:
1955:
1951:
1930:
1910:
1907:
1902:
1898:
1877:
1869:
1850:
1847:
1842:
1838:
1831:
1826:
1822:
1813:
1795:
1791:
1765:
1762:
1759:
1751:
1748:
1745:
1737:
1733:
1729:
1721:
1717:
1713:
1708:
1704:
1700:
1697:
1694:
1689:
1685:
1681:
1676:
1672:
1668:
1663:
1659:
1655:
1650:
1646:
1639:
1632:
1631:
1630:
1628:
1624:
1620:
1616:
1612:
1591:
1588:
1585:
1577:
1574:
1571:
1563:
1559:
1555:
1544:
1540:
1536:
1533:
1530:
1525:
1521:
1517:
1512:
1508:
1498:
1491:
1490:
1489:
1475:
1472:
1469:
1466:
1463:
1460:
1457:
1432:
1428:
1424:
1421:
1416:
1412:
1408:
1403:
1399:
1386:
1384:
1364:
1361:
1358:
1355:
1349:
1346:
1343:
1337:
1334:
1328:
1322:
1319:
1299:
1296:
1290:
1287:
1284:
1278:
1275:
1269:
1263:
1260:
1253:
1252:
1251:
1249:
1246:†1, and its
1245:
1241:
1237:
1233:
1230:
1202:
1199:
1196:
1193:
1190:
1184:
1181:
1149:
1146:
1143:
1140:
1137:
1131:
1128:
1120:
1101:
1098:
1095:
1092:
1089:
1074:
1072:
1024:
1010:
1009:Borel algebra
1006:
1005:sigma algebra
1002:
1001:cylinder sets
998:
994:
990:
986:
982:
978:
974:
950:
946:
911:
908:
905:
899:
888:
884:
858:
855:
852:
846:
843:
836:of copies of
835:
832:
829:Consider the
825:Borel algebra
824:
822:
809:
803:
800:
797:
791:
788:
780:
772:
770:
768:
767:waiting times
760:
756:
752:
749:
745:
741:
737:
733:
730:
726:
722:
718:
714:
713:
712:
710:
706:
701:
699:
694:
690:
683:
679:
672:
665:
658:
653:
651:
646:
642:
637:
635:
630:
626:
618:
616:
613:
611:
607:
603:
599:
594:
592:
589:
580:
576:
572:
556:
548:
544:
540:
536:
532:
531:
530:
525:
518:
511:
508:
505:
501:
493:
491:
489:
484:
482:
478:
474:
469:
465:
461:
460:coin flipping
457:
452:
448:
445:
441:
438:, so it is a
437:
433:
430:(named after
429:
425:
421:
409:
404:
402:
397:
395:
390:
389:
387:
386:
381:
378:
376:
373:
372:
371:
370:
365:
362:
360:
357:
355:
352:
350:
347:
345:
342:
340:
337:
336:
335:
334:
329:
326:
324:
321:
319:
316:
314:
311:
310:
309:
308:
301:
298:
296:
293:
291:
288:
286:
283:
281:
278:
276:
273:
271:
268:
266:
263:
262:
261:
258:
256:
253:
249:
246:
244:
241:
239:
236:
234:
231:
229:
226:
224:
221:
220:
219:
216:
212:
209:
208:
207:
204:
200:
197:
195:
192:
190:
187:
185:
182:
180:
177:
176:
175:
172:
170:
167:
165:
162:
161:
160:
159:
154:
151:
149:
148:Indeterminism
146:
142:
139:
138:
137:
134:
130:
127:
126:
125:
122:
121:
120:
119:
115:
111:
110:
107:
103:
100:
96:
95:
86:
83:
75:
65:
61:
55:
54:
48:
43:
34:
33:
27:
19:
7982:Econometrics
7944:Wiener space
7832:ItĂŽ integral
7733:Inequalities
7622:Self-similar
7592:GaussâMarkov
7582:Exchangeable
7562:CĂ dlĂ g paths
7498:Risk process
7450:LIBOR market
7319:Random graph
7314:Random field
7126:Superprocess
7064:LĂ©vy process
7059:Jump process
7034:Hunt process
6870:Markov chain
6844:
6771:
6734:
6727:
6716:. Retrieved
6696:
6677:
6673:
6644:
6636:
6632:
6628:
6623:
6604:
6598:
6579:
6556:
6548:
6539:
6535:
6533:
6530:
6385:
6383:
6380:
6367:
6357:
6341:
6319:
6295:
6291:
6287:
6272:
6266:January 2014
6263:
6244:that do not
6240:may contain
6239:
6207:
6202:
6198:
6194:
6190:
6187:
6179:uncorrelated
6174:
6170:
6166:
6162:
6160:
6155:
6151:
6147:
6143:
6141:
6094:
6083:
6065:
6063:
6013:
5959:
5954:
5950:
5946:
5944:
5861:of integers
5854:
5852:
5843:
5841:
5771:
5590:is given by
5569:
5550:
5487:
5407:
5226:
5118:
4934:
4704:homomorphism
4629:
4500:
4451:
4402:
4337:
4080:
3770:
3649:
3641:Haar measure
3634:
3535:
3381:
3343:
3311:
3306:
3302:
3298:
3292:
3167:
3163:
3159:
3072:
3066:
3057:
3053:
3048:
3044:
3038:
3025:
3021:
3017:
3015:
2848:
2843:
2839:
2835:
2831:
2829:
2761:
2637:
2633:
2629:
2627:
2515:
2511:
2507:
2503:
2499:
2495:
2487:
2483:
2479:
2477:
2443:
2332:
2331:of flipping
2158:
2094:
2086:measure zero
2002:
1782:
1626:
1622:
1618:
1614:
1610:
1608:
1387:
1382:
1380:
1243:
1242:, where 0 â€
1239:
1235:
1231:
1078:
996:
992:
988:
984:
828:
776:
764:
758:
747:
743:
735:
728:
724:
716:
702:
697:
692:
688:
681:
677:
670:
663:
656:
654:
644:
640:
638:
636:th "trial".
633:
628:
624:
622:
614:
609:
605:
601:
595:
585:
578:
574:
570:
542:
538:
534:
523:
516:
509:
499:
497:
485:
467:
463:
450:
446:
443:
427:
417:
380:Tree diagram
375:Venn diagram
339:Independence
285:Markov chain
264:
169:Sample space
78:
69:
50:
26:
8027:Ruin theory
7965:Disciplines
7837:ItĂŽ's lemma
7612:Predictable
7287:Percolation
7270:Potts model
7265:Ising model
7229:White noise
7187:Differences
7049:ItĂŽ process
6989:Cox process
6885:Loop-erased
6880:Random walk
6394:step number
6152:(1)(0)(1)()
5945:called the
5848:realization
4930:Baker's map
3771:defined by
3728:pushforward
3643:; it is an
3384:shift space
3358:shift space
1783:where each
757:NB(1,
504:independent
420:probability
295:Random walk
136:Determinism
124:Probability
64:introducing
8037:Statistics
7817:Filtration
7718:Kolmogorov
7702:Blumenthal
7627:Stationary
7567:Continuous
7555:Properties
7440:HullâWhite
7182:Martingale
7069:Local time
6957:Fractional
6935:pure birth
6718:2018-07-28
6562:References
6540:1111000111
6210:pseudocode
6015:Almost all
5557:carry bits
5535:Cantor set
5488:gives the
5227:where the
5154:given by
4702:induces a
4682:The shift
3566:, one has
3317:isomorphic
2494:of length
2052:, for any
598:memoryless
494:Definition
424:statistics
206:Experiment
153:Randomness
99:statistics
47:references
7949:Classical
6962:Geometric
6952:Excursion
6454:(0)(1)(1)
6256:. Please
6254:talk page
6242:citations
5893:∈
5842:The term
5749:…
5693:…
5671:…
5615:…
5441:∞
5426:∑
5364:−
5197:−
4898:Ω
4817:⋯
4779:⋯
4667:≤
4661:≤
4580:∞
4565:∑
4535:⋯
4465:↦
4061:−
4053:∘
4002:→
3961:→
3875:→
3842:σ
3831:−
3811:σ
3795:−
3787:∘
3751:−
3743:∘
3709:→
3671:→
3617:σ
3599:σ
3588:−
3547:∈
3544:σ
3518:⋯
3480:⋯
3394:Ω
3329:universal
3272:−
3263:
3244:−
3235:−
3229:
3213:−
3184:∞
3181:→
3143:≤
2955:−
2931:π
2890:∞
2887:→
2784:∑
2739:−
2725:−
2640:heads is
2601:−
2284:∑
2258:¯
2106:Ω
2063:≤
2022:∞
2019:→
1952:ω
1899:ω
1839:ω
1763:−
1749:−
1698:⋯
1589:−
1575:−
1541:ω
1534:⋯
1522:ω
1509:ω
1470:⋯
1450:at times
1429:ω
1425:⋯
1413:ω
1400:ω
1362:−
1200:−
1147:−
1099:−
1022:Ω
944:Ω
882:Ω
533:for each
199:Singleton
8085:Category
8070:Category
7954:Abstract
7488:BĂŒhlmann
7094:Compound
6751:Archived
6709:Archived
6465:(10)(11)
6457:(1)()()
6451:()(0)(1)
6248:the text
6144:10011011
6136:discard
6112:discard
5859:sequence
5553:odometer
5541:Odometer
5254:are the
4450:The map
4342:spectrum
4081:The map
3362:odometer
3047:exactly
973:topology
280:Variance
7577:Ergodic
7465:VaĆĄĂÄek
7307:Poisson
6967:Meander
6104:output
4336:or the
3069:entropy
2762:where
2492:strings
1174:(or by
1119:measure
981:strings
746:,
727:,
662:,
522:,
515:,
194:Outcome
60:improve
7917:Tanaka
7602:Mixing
7597:Markov
7470:Wilkie
7435:HoâLee
7430:Heston
7202:Super-
6947:Bridge
6895:Biased
6778:
6652:
6611:
6586:
6482:(11)()
6474:()(1)
6471:(1)(0)
6400:output
6305:output
6246:verify
4547:write
3726:. The
3333:mixing
2239:. The
2153:, and
1609:where
1234:has a
141:System
129:Axioms
49:, but
7770:Tools
7546:M/M/c
7541:M/M/1
7536:M/G/1
7526:Fluid
7192:Local
6754:(PDF)
6739:(PDF)
6712:(PDF)
6705:(PDF)
6397:input
6302:input
6101:input
5955:heads
5826:is a
4112:is a
3639:is a
3307:heads
2636:with
2333:heads
1814:with
573:that
174:Event
7722:LĂ©vy
7521:Bulk
7405:Chen
7197:Sub-
7155:Both
6776:ISBN
6650:ISBN
6609:ISBN
6584:ISBN
6499:(10)
6491:(1)
6368:none
6358:none
6342:none
6320:none
6046:are
4214:and
3376:and
3132:for
3056:and
2199:and
2069:<
1385:).
1312:and
987:and
687:and
454:are
426:, a
422:and
7302:Cox
6743:doi
6682:doi
6525:()
6508:()
6505:(1)
6502:(1)
6488:(0)
6468:(1)
6165:(1â
6158:).
6156:101
4865:mod
4475:mod
3254:log
3220:log
2012:lim
1943:if
1923:or
1890:if
1866:in
983:of
742:NB(
612:.)
418:In
8087::
7720:,
7716:,
7712:,
7708:,
7704:,
6749:.
6678:20
6676:.
6672:.
6660:^
6637:25
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6570:^
6546:.
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6032:x
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5926:1
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5920:)
5917:x
5914:(
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5901::
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5890:n
5887:{
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5466:+
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5433:=
5430:n
5422:=
5419:y
5388:)
5385:y
5382:(
5377:n
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5367:n
5360:2
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5343:1
5340:+
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5325:n
5321:B
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5312:1
5307:+
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5284:n
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5274:2
5271:1
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5210:n
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5189:=
5184:n
5180:B
5174:T
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5098:2
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4737:(
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4300:g
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4147:g
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2678:]
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2659:[
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2016:n
1984:P
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1961:=
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1911:H
1908:=
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807:}
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