1569:
application. Athreya identifies the three classes of size-dependent branching processes as sub-critical, stable, and super-critical branching measures. For
Athreya, the central parameters are crucial to control if sub-critical and super-critical unstable branching is to be avoided. Size dependent branching processes are also discussed under the topic of
1560:) for the number of offspring of a given parent, if the mean number of offspring produced by a single parent is less than or equal to one, then the ultimate extinction probability is one. If the mean number of offspring produced by a single parent is greater than one, then the ultimate extinction probability is strictly less than one.
1472:
92:
that does not vary from individual to individual. Branching processes are used to model reproduction; for example, the individuals might correspond to bacteria, each of which generates 0, 1, or 2 offspring with some probability in a single time unit. Branching processes can also
2304:
There are many other branching processes, for example, branching processes in random environments, in which the reproduction law is chosen randomly at each generation, or branching processes, where the growth of the population is controlled by external influences or interacting processes. Branching
1959:
Branching processes can be simulated for a range of problems. One specific use of simulated branching process is in the field of evolutionary biology. Phylogenetic trees, for example, can be simulated under several models, helping to develop and validate estimation methods as well as supporting
546:
for all individuals. For continuous-time branching processes, each individual waits for a random time (which is a continuous random variable), and then divides according to the given distribution. The waiting time for different individuals are independent, and are independent with the number of
1568:
Along with discussion of a more general model of branching processes known as age-dependent branching processes by
Grimmett, in which individuals live for more than one generation, Krishna Athreya has identified three distinctions between size-dependent branching processes which have general
533:
is visited. Then in each period, the number of revealed but unvisited nodes equals the number of such nodes in the previous period, plus the new nodes that are revealed when visiting a node, minus the node that is visited. The process ends once all revealed nodes have been visited.
2285:
For multitype branching processes that the populations of different types grow exponentially, the proportions of different types converge almost surely to a constant vector under some mild conditions. This is the strong law of large numbers for multitype branching processes.
630:
For any nontrivial cases (trivial cases are ones in which the probability of having no offspring is zero for every member of the population - in such cases the probability of ultimate extinction is 0), the probability of ultimate extinction equals one if
2305:
processes where particles have to work (contribute resources to the environment) in order to be able to reproduce, and live in a changing society structure controlling the distribution of resources, are so-called resource-dependent branching processes.
967:
507:= 1. To gain some intuition for this formulation, imagine a walk where the goal is to visit every node, but every time a previously unvisited node is visited, additional nodes are revealed that must also be visited. Let
1466:
1359:
42:
indexed by some set, usually natural or non-negative real numbers. The original purpose of branching processes was to serve as a mathematical model of a population in which each individual in generation
495:
148:> 1, then the probability of ultimate extinction is less than 1 (but not necessarily zero; consider a process where each individual either has 0 or 100 children with equal probability. In that case,
1696:
763:
1081:
2296:
The monograph by
Athreya and Ney summarizes a common set of conditions under which this law of large numbers is valid. Later there are some improvements through discarding different conditions.
328:
625:
2793:, 2nd ed., Clarendon Press, Oxford, 1992. Section 5.4 discusses the model of branching processes described above. Section 5.5 discusses a more general model of branching processes known as
2867:
227:
2175:
1146:
3402:
2032:
2003:
2275:
1208:
3226:
2779:
2229:
2202:
2116:
2089:
2062:
3829:
2822:
86:
3359:
3339:
60:
809:
3743:
2037:
For example, consider the population of cancer stem cells (CSCs) and non-stem cancer cells (NSCCs). After each time interval, each CSC has probability
3660:
2293:
system, which has a unique attracting fixed point. This fixed point is just the vector that the proportions converge to in the law of large numbers.
3670:
3344:
1468:) function. There are at most two intersection points. Since (1,1) is always an intersect point for the two functions, there only exist three cases:
3354:
3712:
3427:
3609:
1520:
In case 1, the ultimate extinction probability is strictly less than one. For case 2 and 3, the ultimate extinction probability equals to one.
1552:
is exactly the expected number of offspring a parent could produce, it can be concluded that for a branching process with generating function
152:= 50, but probability of ultimate extinction is greater than 0.5, since that's the probability that the first individual has 0 children). If
3899:
3889:
3412:
2669:
2548:"TreeSimGM: Simulating phylogenetic trees under general BellmanâHarris models with lineage-specific shifts of speciation and extinction in R"
3799:
3763:
2428:
and M. Duerinckx (2015) "Resource dependent branching processes and the envelope of societies", Annals of
Applied Probability. 25: 324â372.
2336:
1570:
3716:
4067:
3804:
2914:
2815:
2388:
1364:
383:
3869:
3447:
3417:
1586:
697:
2786:, 2nd ed. Section 10.3 discusses branching processes in detail together with the application of generating functions to study them.
1260:
3720:
3704:
3914:
3619:
2839:
3819:
3784:
3753:
3748:
3387:
3184:
3101:
2346:
643:
3758:
3086:
667:, ... be the probabilities of producing 0, 1, 2, ... offspring by each individual in each generation. Let
3382:
3189:
2290:
995:
3108:
3844:
3724:
2776:
4072:
3849:
3685:
3584:
3569:
2981:
2897:
2808:
3859:
3495:
3854:
253:
156: = 1, then ultimate extinction occurs with probability 1 unless each individual always has exactly one child.
3457:
2341:
569:
3041:
2986:
2902:
3789:
3779:
3422:
3392:
3794:
2959:
2857:
2005:, a random vector representing the numbers of children in different types, satisfies a probability distribution on
1951: = 1/3, and this is the value to which the extinction probability converges with increasing generations.
3505:
3081:
2862:
2321:
180:
4093:
3874:
3675:
3589:
3574:
2964:
20:
3708:
3594:
3016:
2605:"The overshoot and phenotypic equilibrium in characterizing cancer dynamics of reversible phenotypic plasticity"
3096:
3071:
89:
3814:
3397:
2932:
789:
offspring in the first generation, then to die out by the mth generation, each of these lines must die out in
2403:
G. R. Grimmett and D. R. Stirzaker, Probability and Random
Processes, 2nd ed., Clarendon Press, Oxford, 1992.
4009:
3999:
3690:
3472:
3211:
3076:
2887:
168:
3294:
3951:
3879:
3138:
2604:
560:
141:
2495:
Hagen, Oskar; Andermann, Tobias; Quental, Tiago B.; Antonelli, Alexandre; Silvestro, Daniele (May 2018).
2121:
1092:
3974:
3956:
3936:
3931:
3650:
3482:
3462:
3309:
3252:
3091:
3001:
2331:
136:< 1, then the expected number of individuals goes rapidly to zero, which implies ultimate extinction
3442:
1580:
Consider a parent can produce at most two offspring. The extinction probability in each generation is:
2008:
1979:
4049:
4004:
3994:
3735:
3680:
3655:
3624:
3604:
3364:
3349:
3216:
2731:
4044:
3884:
3809:
3614:
3374:
3284:
3174:
2234:
160:
691:
generation must be added up, the extinction probability is nondecreasing in generations. That is,
4014:
3979:
3894:
3864:
3634:
3629:
3452:
3289:
2954:
2892:
2831:
2720:"Phenotypic equilibrium as probabilistic convergence in multi-phenotype cell population dynamics"
2642:
2616:
137:
35:
27:
3695:
113:
1174:
4034:
3839:
3490:
3247:
3133:
3026:
3006:
2996:
2847:
2759:
2665:
2634:
2585:
2567:
2528:
2477:
2459:
2384:
3700:
3437:
4054:
3941:
3824:
3194:
3169:
3118:
3046:
2969:
2922:
2749:
2739:
2698:
2626:
2575:
2559:
2518:
2508:
2467:
2451:
2376:
1968:
In multitype branching processes, individuals are not identical, but can be classified into
39:
2207:
2180:
2094:
2067:
2040:
4019:
3919:
3904:
3665:
3599:
3277:
3221:
3204:
2949:
2783:
2425:
102:
3834:
3066:
1764: = 0.3, the extinction probability for the first 20 generations is as follows:
962:{\displaystyle d_{m}=p_{0}+p_{1}d_{m-1}+p_{2}(d_{m-1})^{2}+p_{3}(d_{m-1})^{3}+\cdots .\,}
65:
2735:
2687:"Functional limit theorems for multitype branching processes and generalized PĂłlya urns"
2497:"Estimating Age-Dependent Extinction: Contrasting Evidence from Fossils and Phylogenies"
4024:
3989:
3909:
3515:
3262:
3179:
3148:
3143:
3123:
3113:
3056:
3051:
3031:
3011:
2976:
2944:
2927:
2754:
2719:
2580:
2547:
2523:
2496:
2472:
2439:
117:
45:
4087:
3926:
3467:
3304:
3299:
3257:
3199:
3021:
2937:
2877:
2646:
2603:
Chen, Xiufang; Wang, Yue; Feng, Tianquan; Yi, Ming; Zhang, Xingan; Zhou, Da (2016).
3984:
3946:
3500:
3432:
3321:
3316:
3128:
3061:
3036:
2872:
2351:
2309:
3564:
793: â 1 generations. Since they proceed independently, the probability is (
2744:
2380:
547:
children. In general, the waiting time is an exponential variable with parameter
4029:
3548:
3543:
3538:
3528:
3331:
3272:
3267:
3231:
2991:
2882:
2326:
2289:
For continuous-time cases, proportions of the population expectation satisfy an
344:
2630:
4039:
3579:
3523:
3407:
2703:
2686:
972:
The right-hand side of the equation is a probability generating function. Let
108:
A central question in the theory of branching processes is the probability of
2571:
2463:
2308:
The scaling limit of near-critical branching processes can be used to obtain
112:, where no individuals exist after some finite number of generations. Using
3533:
2563:
2513:
2455:
116:, it can be shown that starting with one individual in generation zero, the
98:
2763:
2638:
2589:
2532:
2481:
542:
For discrete-time branching processes, the "branching time" is fixed to be
93:
be used to model other systems with similar dynamics, e.g., the spread of
2440:"Age-Dependent Speciation Can Explain the Shape of Empirical Phylogenies"
1708: = 0. For the ultimate extinction probability, we need to find
209:
be a random variable denoting the number of direct successors of member
3360:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
2800:
94:
1743:
Taking as example probabilities for the numbers of offspring produced
1213:
This is also equivalent to finding the intersection point(s) of lines
2621:
2091:
to produce one CSC and one NSCC (asymmetric division), probability
1471:
1470:
179:
The most common formulation of a branching process is that of the
687:= 0. Since the probabilities for all paths that lead to 0 by the
1461:{\displaystyle h''(z)=2p_{2}+6p_{3}z+12p_{4}z^{2}+\cdots \geq 0}
490:{\displaystyle S_{i+1}=S_{i}+X_{i+1}-1=\sum _{j=1}^{i+1}X_{j}-i}
2804:
516:
represent the number of revealed but unvisited nodes in period
2438:
Hagen, O.; Hartmann, K.; Steel, M.; Stadler, T. (2015-05-01).
1691:{\displaystyle d_{m}=p_{0}+p_{1}d_{m-1}+p_{2}(d_{m-1})^{2}.\,}
758:{\displaystyle 0=d_{0}\leq d_{1}\leq d_{2}\leq \cdots \leq 1.}
529:
represent the number of new nodes that are revealed when node
370:
62:
produces some random number of individuals in generation
1354:{\displaystyle h'(z)=p_{1}+2p_{2}z+3p_{3}z^{2}+\cdots \geq 0}
1086:
Using the generating function, the previous equation becomes
343:
Alternatively, the branching process can be formulated as a
132:
is the expected number of children of each individual. If
3340:
Autoregressive conditional heteroskedasticity (ARCH) model
21:
Restricted representation § Classical branching rules
2797:, in which individuals live for more than one generation.
2868:
Independent and identically distributed random variables
228:
independent and identically distributed random variables
2204:
to produce two NSCCs (symmetric division), probability
1076:{\displaystyle h(z)=p_{0}+p_{1}z+p_{2}z^{2}+\cdots .\,}
551:
for all individuals, so that the process is
Markovian.
3345:
Autoregressive integrated moving average (ARIMA) model
2281:
Law of large numbers for multitype branching processes
2177:
to produce nothing (death); each NSCC has probability
2064:
to produce two CSCs (symmetric division), probability
2237:
2210:
2183:
2124:
2097:
2070:
2043:
2011:
1982:
1589:
1367:
1263:
1177:
1095:
998:
812:
785:
is the ultimate extinction probability. If there are
700:
572:
386:
256:
68:
48:
3967:
3772:
3734:
3643:
3557:
3514:
3481:
3373:
3330:
3240:
3157:
2913:
2838:
1972:types. After each time step, an individual of type
2269:
2231:to produce one NSCC (stagnation), and probability
2223:
2196:
2169:
2110:
2083:
2056:
2026:
1997:
1690:
1460:
1353:
1202:
1140:
1075:
961:
757:
619:
489:
323:{\displaystyle Z_{n+1}=\sum _{i=1}^{Z_{n}}X_{n,i}}
322:
196:(often interpreted as the size of generation
80:
54:
2118:to produce one CSC (stagnation), and probability
1976:will produce individuals of different types, and
1947:In this example, we can solve algebraically that
620:{\displaystyle \lim _{n\to \infty }\Pr(Z_{n}=0).}
3227:Stochastic chains with memory of variable length
642:The process can be analyzed using the method of
589:
574:
234: â{ 0, 1, 2, ...} and
555:Extinction problem for a GaltonâWatson process
88:, according, in the simplest case, to a fixed
19:For the process in representation theory, see
2816:
2664:. Berlin: Springer-Verlag. pp. 199â206.
635: †1 and strictly less than one if
8:
2718:Jiang, Da-Quan; Wang, Yue; Zhou, Da (2017).
2371:Athreya, K. B. (2006). "Branching Process".
34:is a type of mathematical object known as a
2691:Stochastic Processes and Their Applications
2660:Athreya, Krishna B.; Ney, Peter E. (1972).
3355:Autoregressiveâmoving-average (ARMA) model
2823:
2809:
2801:
1517:> 1.(See the black curve in the graph)
980:) be the ordinary generating function for
2753:
2743:
2702:
2620:
2579:
2522:
2512:
2471:
2261:
2248:
2236:
2215:
2209:
2188:
2182:
2161:
2148:
2135:
2123:
2102:
2096:
2075:
2069:
2048:
2042:
2018:
2014:
2013:
2010:
1989:
1984:
1981:
1687:
1678:
1662:
1649:
1630:
1620:
1607:
1594:
1588:
1503:< 1 (see the red curve in the graph).
1440:
1430:
1411:
1395:
1366:
1333:
1323:
1304:
1288:
1262:
1199:
1176:
1137:
1119:
1100:
1094:
1072:
1057:
1047:
1031:
1018:
997:
958:
943:
927:
914:
901:
885:
872:
853:
843:
830:
817:
811:
737:
724:
711:
699:
599:
577:
571:
475:
459:
448:
423:
410:
391:
385:
308:
296:
291:
280:
261:
255:
67:
47:
1766:
2363:
1510:= 1.(See the green curve in the graph)
1506:Case 2 has only one intersect point at
3661:Doob's martingale convergence theorems
1513:Case 3 has another intersect point at
1499:Case 1 has another intersect point at
3413:Constant elasticity of variance (CEV)
3403:ChanâKarolyiâLongstaffâSanders (CKLS)
2546:Hagen, Oskar; Stadler, Tanja (2018).
676:be the extinction probability by the
167:of a branching process is called the
7:
2789:G. R. Grimmett and D. R. Stirzaker,
2337:Resource-dependent branching process
1571:resource-dependent branching process
247:}. Then the recurrence equation is
101:or the propagation of neutrons in a
2170:{\displaystyle 1-p_{1}-p_{2}-p_{3}}
1141:{\displaystyle d_{m}=h(d_{m-1}).\,}
538:Continuous-time branching processes
377:. Then the recurrence equation is
38:, which consists of collections of
3900:Skorokhod's representation theorem
3681:Law of large numbers (weak/strong)
2412:Krishna Athreya and Peter Jagers.
1564:Size dependent branching processes
584:
14:
3870:Martingale representation theorem
2795:age-dependent branching processes
2775:C. M. Grinstead and J. L. Snell,
3915:Stochastic differential equation
3805:Doob's optional stopping theorem
3800:DoobâMeyer decomposition theorem
2791:Probability and Random Processes
2552:Methods in Ecology and Evolution
2027:{\displaystyle \mathbb {N} ^{n}}
1998:{\displaystyle \mathbf {X} _{i}}
1985:
238: â {1, ...,
3785:Convergence of random variables
3671:FisherâTippettâGnedenko theorem
2347:Martingale (probability theory)
644:probability generating function
3383:Binomial options pricing model
2609:Journal of Theoretical Biology
2373:Encyclopedia of Environmetrics
1955:Simulating branching processes
1675:
1655:
1382:
1376:
1278:
1272:
1193:
1187:
1131:
1112:
1008:
1002:
940:
920:
898:
878:
611:
592:
581:
1:
3850:Kolmogorov continuity theorem
3686:Law of the iterated logarithm
2270:{\displaystyle 1-p_{4}-p_{5}}
1964:Multitype branching processes
1576:Example of extinction problem
1548: + ... =
369:be a random variable that is
3855:Kolmogorov extension theorem
3534:Generalized queueing network
3042:Interacting particle systems
2745:10.1371/journal.pone.0170916
2381:10.1002/9780470057339.vab032
2277:to produce nothing (death).
2987:Continuous-time random walk
2777:Introduction to Probability
2322:Galton–Watson process
356:denote the state in period
192:denote the state in period
181:Galton–Watson process
4110:
3995:Extreme value theory (EVT)
3795:Doob decomposition theorem
3087:OrnsteinâUhlenbeck process
2858:Chinese restaurant process
2631:10.1016/j.jtbi.2015.11.008
1257:) is an increasing (since
18:
16:Kind of stochastic process
4063:
3875:Optional stopping theorem
3676:Large deviation principle
3428:HeathâJarrowâMorton (HJM)
3365:Moving-average (MA) model
3350:Autoregressive (AR) model
3175:Hidden Markov model (HMM)
3109:SchrammâLoewner evolution
2704:10.1016/j.spa.2003.12.002
2300:Other branching processes
1203:{\displaystyle d=h(d).\,}
3790:Doléans-Dade exponential
3620:Progressively measurable
3418:CoxâIngersollâRoss (CIR)
1168:can be found by solving
175:Mathematical formulation
120:size of generation
90:probability distribution
4010:Mathematical statistics
4000:Large deviations theory
3830:Infinitesimal generator
3691:Maximal ergodic theorem
3610:Piecewise-deterministic
3212:Random dynamical system
3077:Markov additive process
2685:Janson, Svante (2003).
2564:10.1111/2041-210X.12917
2342:BrussâDuerinckx theorem
1781:Extinction probability
680:generation. Obviously,
169:basic reproductive rate
3845:KarhunenâLoĂšve theorem
3780:CameronâMartin formula
3744:BurkholderâDavisâGundy
3139:Variance gamma process
2271:
2225:
2198:
2171:
2112:
2085:
2058:
2028:
1999:
1773:Extinction probability
1757: = 0.6, and
1692:
1496:
1462:
1355:
1204:
1142:
1077:
963:
759:
621:
561:extinction probability
491:
470:
324:
303:
82:
56:
3975:Actuarial mathematics
3937:Uniform integrability
3932:Stratonovich integral
3860:LĂ©vyâProkhorov metric
3764:MarcinkiewiczâZygmund
3651:Central limit theorem
3253:Gaussian random field
3082:McKeanâVlasov process
3002:Dyson Brownian motion
2863:GaltonâWatson process
2514:10.1093/sysbio/syx082
2456:10.1093/sysbio/syv001
2332:Branching random walk
2272:
2226:
2224:{\displaystyle p_{5}}
2199:
2197:{\displaystyle p_{4}}
2172:
2113:
2111:{\displaystyle p_{3}}
2086:
2084:{\displaystyle p_{2}}
2059:
2057:{\displaystyle p_{1}}
2029:
2000:
1693:
1474:
1463:
1356:
1205:
1143:
1078:
964:
777:converges to a limit
760:
622:
492:
444:
325:
276:
144:. Alternatively, if
83:
57:
4050:Time series analysis
4005:Mathematical finance
3890:Reflection principle
3217:Regenerative process
3017:FlemingâViot process
2832:Stochastic processes
2235:
2208:
2181:
2122:
2095:
2068:
2041:
2009:
1980:
1960:hypothesis testing.
1778:Generation # (11â20)
1587:
1365:
1361:) and convex (since
1261:
1245:is a straight line.
1175:
1093:
996:
810:
698:
570:
384:
254:
66:
46:
4045:Stochastic analysis
3885:Quadratic variation
3880:Prokhorov's theorem
3815:FeynmanâKac formula
3285:Markov random field
2933:Birthâdeath process
2736:2017PLoSO..1270916J
2662:Branching Processes
2414:Branching Processes
1770:Generation # (1â10)
639: > 1.
161:theoretical ecology
142:Markov's inequality
110:ultimate extinction
81:{\displaystyle n+1}
4015:Probability theory
3895:Skorokhod integral
3865:Malliavin calculus
3448:Korn-Kreer-Lenssen
3332:Time series models
3295:PitmanâYor process
2782:2011-07-27 at the
2501:Systematic Biology
2444:Systematic Biology
2267:
2221:
2194:
2167:
2108:
2081:
2054:
2024:
1995:
1750: = 0.1,
1688:
1523:By observing that
1497:
1458:
1351:
1200:
1138:
1073:
959:
755:
617:
588:
487:
320:
138:with probability 1
78:
52:
36:stochastic process
28:probability theory
4081:
4080:
4035:Signal processing
3754:Doob's upcrossing
3749:Doob's martingale
3713:EngelbertâSchmidt
3656:Donsker's theorem
3590:Feller-continuous
3458:RendlemanâBartter
3248:Dirichlet process
3165:Branching process
3134:Telegraph process
3027:Geometric process
3007:Empirical process
2997:Diffusion process
2853:Branching process
2848:Bernoulli process
2671:978-3-642-65371-1
2416:. Springer. 1973.
1945:
1944:
1487:) intersect with
573:
55:{\displaystyle n}
32:branching process
4101:
4094:Markov processes
4055:Machine learning
3942:Usual hypotheses
3825:Girsanov theorem
3810:Dynkin's formula
3575:Continuous paths
3483:Actuarial models
3423:GarmanâKohlhagen
3393:BlackâKarasinski
3388:BlackâDermanâToy
3375:Financial models
3241:Fields and other
3170:Gaussian process
3119:Sigma-martingale
2923:Additive process
2825:
2818:
2811:
2802:
2768:
2767:
2757:
2747:
2715:
2709:
2708:
2706:
2682:
2676:
2675:
2657:
2651:
2650:
2624:
2600:
2594:
2593:
2583:
2543:
2537:
2536:
2526:
2516:
2492:
2486:
2485:
2475:
2435:
2429:
2423:
2417:
2410:
2404:
2401:
2395:
2394:
2368:
2276:
2274:
2273:
2268:
2266:
2265:
2253:
2252:
2230:
2228:
2227:
2222:
2220:
2219:
2203:
2201:
2200:
2195:
2193:
2192:
2176:
2174:
2173:
2168:
2166:
2165:
2153:
2152:
2140:
2139:
2117:
2115:
2114:
2109:
2107:
2106:
2090:
2088:
2087:
2082:
2080:
2079:
2063:
2061:
2060:
2055:
2053:
2052:
2033:
2031:
2030:
2025:
2023:
2022:
2017:
2004:
2002:
2001:
1996:
1994:
1993:
1988:
1767:
1712:which satisfies
1697:
1695:
1694:
1689:
1683:
1682:
1673:
1672:
1654:
1653:
1641:
1640:
1625:
1624:
1612:
1611:
1599:
1598:
1527:(1) =
1467:
1465:
1464:
1459:
1445:
1444:
1435:
1434:
1416:
1415:
1400:
1399:
1375:
1360:
1358:
1357:
1352:
1338:
1337:
1328:
1327:
1309:
1308:
1293:
1292:
1271:
1237: â„ 0.
1209:
1207:
1206:
1201:
1147:
1145:
1144:
1139:
1130:
1129:
1105:
1104:
1082:
1080:
1079:
1074:
1062:
1061:
1052:
1051:
1036:
1035:
1023:
1022:
968:
966:
965:
960:
948:
947:
938:
937:
919:
918:
906:
905:
896:
895:
877:
876:
864:
863:
848:
847:
835:
834:
822:
821:
764:
762:
761:
756:
742:
741:
729:
728:
716:
715:
626:
624:
623:
618:
604:
603:
587:
496:
494:
493:
488:
480:
479:
469:
458:
434:
433:
415:
414:
402:
401:
329:
327:
326:
321:
319:
318:
302:
301:
300:
290:
272:
271:
163:, the parameter
87:
85:
84:
79:
61:
59:
58:
53:
40:random variables
4109:
4108:
4104:
4103:
4102:
4100:
4099:
4098:
4084:
4083:
4082:
4077:
4059:
4020:Queueing theory
3963:
3905:Skorokhod space
3768:
3759:KunitaâWatanabe
3730:
3696:Sanov's theorem
3666:Ergodic theorem
3639:
3635:Time-reversible
3553:
3516:Queueing models
3510:
3506:SparreâAnderson
3496:CramĂ©râLundberg
3477:
3463:SABR volatility
3369:
3326:
3278:Boolean network
3236:
3222:Renewal process
3153:
3102:Non-homogeneous
3092:Poisson process
2982:Contact process
2945:Brownian motion
2915:Continuous time
2909:
2903:Maximal entropy
2834:
2829:
2784:Wayback Machine
2772:
2771:
2730:(2): e0170916.
2717:
2716:
2712:
2684:
2683:
2679:
2672:
2659:
2658:
2654:
2602:
2601:
2597:
2545:
2544:
2540:
2494:
2493:
2489:
2437:
2436:
2432:
2426:F. Thomas Bruss
2424:
2420:
2411:
2407:
2402:
2398:
2391:
2370:
2369:
2365:
2360:
2318:
2302:
2283:
2257:
2244:
2233:
2232:
2211:
2206:
2205:
2184:
2179:
2178:
2157:
2144:
2131:
2120:
2119:
2098:
2093:
2092:
2071:
2066:
2065:
2044:
2039:
2038:
2012:
2007:
2006:
1983:
1978:
1977:
1966:
1957:
1763:
1756:
1749:
1736:
1729:
1722:
1707:
1674:
1658:
1645:
1626:
1616:
1603:
1590:
1585:
1584:
1578:
1566:
1547:
1540:
1533:
1475:Three cases of
1436:
1426:
1407:
1391:
1368:
1363:
1362:
1329:
1319:
1300:
1284:
1264:
1259:
1258:
1173:
1172:
1159:
1115:
1096:
1091:
1090:
1053:
1043:
1027:
1014:
994:
993:
988:
939:
923:
910:
897:
881:
868:
849:
839:
826:
813:
808:
807:
802:
776:
733:
720:
707:
696:
695:
686:
675:
666:
659:
652:
595:
568:
567:
557:
540:
528:
515:
506:
471:
419:
406:
387:
382:
381:
368:
355:
339:
304:
292:
257:
252:
251:
246:
225:
208:
191:
177:
114:Wald's equation
103:nuclear reactor
64:
63:
44:
43:
24:
17:
12:
11:
5:
4107:
4105:
4097:
4096:
4086:
4085:
4079:
4078:
4076:
4075:
4070:
4068:List of topics
4064:
4061:
4060:
4058:
4057:
4052:
4047:
4042:
4037:
4032:
4027:
4025:Renewal theory
4022:
4017:
4012:
4007:
4002:
3997:
3992:
3990:Ergodic theory
3987:
3982:
3980:Control theory
3977:
3971:
3969:
3965:
3964:
3962:
3961:
3960:
3959:
3954:
3944:
3939:
3934:
3929:
3924:
3923:
3922:
3912:
3910:Snell envelope
3907:
3902:
3897:
3892:
3887:
3882:
3877:
3872:
3867:
3862:
3857:
3852:
3847:
3842:
3837:
3832:
3827:
3822:
3817:
3812:
3807:
3802:
3797:
3792:
3787:
3782:
3776:
3774:
3770:
3769:
3767:
3766:
3761:
3756:
3751:
3746:
3740:
3738:
3732:
3731:
3729:
3728:
3709:BorelâCantelli
3698:
3693:
3688:
3683:
3678:
3673:
3668:
3663:
3658:
3653:
3647:
3645:
3644:Limit theorems
3641:
3640:
3638:
3637:
3632:
3627:
3622:
3617:
3612:
3607:
3602:
3597:
3592:
3587:
3582:
3577:
3572:
3567:
3561:
3559:
3555:
3554:
3552:
3551:
3546:
3541:
3536:
3531:
3526:
3520:
3518:
3512:
3511:
3509:
3508:
3503:
3498:
3493:
3487:
3485:
3479:
3478:
3476:
3475:
3470:
3465:
3460:
3455:
3450:
3445:
3440:
3435:
3430:
3425:
3420:
3415:
3410:
3405:
3400:
3395:
3390:
3385:
3379:
3377:
3371:
3370:
3368:
3367:
3362:
3357:
3352:
3347:
3342:
3336:
3334:
3328:
3327:
3325:
3324:
3319:
3314:
3313:
3312:
3307:
3297:
3292:
3287:
3282:
3281:
3280:
3275:
3265:
3263:Hopfield model
3260:
3255:
3250:
3244:
3242:
3238:
3237:
3235:
3234:
3229:
3224:
3219:
3214:
3209:
3208:
3207:
3202:
3197:
3192:
3182:
3180:Markov process
3177:
3172:
3167:
3161:
3159:
3155:
3154:
3152:
3151:
3149:Wiener sausage
3146:
3144:Wiener process
3141:
3136:
3131:
3126:
3124:Stable process
3121:
3116:
3114:Semimartingale
3111:
3106:
3105:
3104:
3099:
3089:
3084:
3079:
3074:
3069:
3064:
3059:
3057:Jump diffusion
3054:
3049:
3044:
3039:
3034:
3032:Hawkes process
3029:
3024:
3019:
3014:
3012:Feller process
3009:
3004:
2999:
2994:
2989:
2984:
2979:
2977:Cauchy process
2974:
2973:
2972:
2967:
2962:
2957:
2952:
2942:
2941:
2940:
2930:
2928:Bessel process
2925:
2919:
2917:
2911:
2910:
2908:
2907:
2906:
2905:
2900:
2895:
2890:
2880:
2875:
2870:
2865:
2860:
2855:
2850:
2844:
2842:
2836:
2835:
2830:
2828:
2827:
2820:
2813:
2805:
2799:
2798:
2787:
2770:
2769:
2710:
2697:(2): 177â245.
2677:
2670:
2652:
2595:
2558:(3): 754â760.
2538:
2507:(3): 458â474.
2487:
2450:(3): 432â440.
2430:
2418:
2405:
2396:
2390:978-0471899976
2389:
2362:
2361:
2359:
2356:
2355:
2354:
2349:
2344:
2339:
2334:
2329:
2324:
2317:
2314:
2310:superprocesses
2301:
2298:
2282:
2279:
2264:
2260:
2256:
2251:
2247:
2243:
2240:
2218:
2214:
2191:
2187:
2164:
2160:
2156:
2151:
2147:
2143:
2138:
2134:
2130:
2127:
2105:
2101:
2078:
2074:
2051:
2047:
2021:
2016:
1992:
1987:
1965:
1962:
1956:
1953:
1943:
1942:
1939:
1936:
1934:
1931:
1927:
1926:
1923:
1920:
1918:
1915:
1911:
1910:
1907:
1904:
1902:
1899:
1895:
1894:
1891:
1888:
1886:
1883:
1879:
1878:
1875:
1872:
1870:
1867:
1863:
1862:
1859:
1856:
1854:
1851:
1847:
1846:
1843:
1840:
1838:
1835:
1831:
1830:
1827:
1824:
1822:
1819:
1815:
1814:
1811:
1808:
1806:
1803:
1799:
1798:
1795:
1792:
1790:
1787:
1783:
1782:
1779:
1776:
1774:
1771:
1761:
1754:
1747:
1734:
1730:d +
1727:
1720:
1705:
1699:
1698:
1686:
1681:
1677:
1671:
1668:
1665:
1661:
1657:
1652:
1648:
1644:
1639:
1636:
1633:
1629:
1623:
1619:
1615:
1610:
1606:
1602:
1597:
1593:
1577:
1574:
1565:
1562:
1545:
1541: + 3
1538:
1534: + 2
1531:
1457:
1454:
1451:
1448:
1443:
1439:
1433:
1429:
1425:
1422:
1419:
1414:
1410:
1406:
1403:
1398:
1394:
1390:
1387:
1384:
1381:
1378:
1374:
1371:
1350:
1347:
1344:
1341:
1336:
1332:
1326:
1322:
1318:
1315:
1312:
1307:
1303:
1299:
1296:
1291:
1287:
1283:
1280:
1277:
1274:
1270:
1267:
1211:
1210:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1155:
1149:
1148:
1136:
1133:
1128:
1125:
1122:
1118:
1114:
1111:
1108:
1103:
1099:
1084:
1083:
1071:
1068:
1065:
1060:
1056:
1050:
1046:
1042:
1039:
1034:
1030:
1026:
1021:
1017:
1013:
1010:
1007:
1004:
1001:
984:
970:
969:
957:
954:
951:
946:
942:
936:
933:
930:
926:
922:
917:
913:
909:
904:
900:
894:
891:
888:
884:
880:
875:
871:
867:
862:
859:
856:
852:
846:
842:
838:
833:
829:
825:
820:
816:
797:
772:
766:
765:
754:
751:
748:
745:
740:
736:
732:
727:
723:
719:
714:
710:
706:
703:
684:
671:
664:
657:
650:
628:
627:
616:
613:
610:
607:
602:
598:
594:
591:
586:
583:
580:
576:
556:
553:
539:
536:
524:
511:
504:
498:
497:
486:
483:
478:
474:
468:
465:
462:
457:
454:
451:
447:
443:
440:
437:
432:
429:
426:
422:
418:
413:
409:
405:
400:
397:
394:
390:
364:
351:
337:
331:
330:
317:
314:
311:
307:
299:
295:
289:
286:
283:
279:
275:
270:
267:
264:
260:
242:
221:
204:
187:
176:
173:
77:
74:
71:
51:
15:
13:
10:
9:
6:
4:
3:
2:
4106:
4095:
4092:
4091:
4089:
4074:
4071:
4069:
4066:
4065:
4062:
4056:
4053:
4051:
4048:
4046:
4043:
4041:
4038:
4036:
4033:
4031:
4028:
4026:
4023:
4021:
4018:
4016:
4013:
4011:
4008:
4006:
4003:
4001:
3998:
3996:
3993:
3991:
3988:
3986:
3983:
3981:
3978:
3976:
3973:
3972:
3970:
3966:
3958:
3955:
3953:
3950:
3949:
3948:
3945:
3943:
3940:
3938:
3935:
3933:
3930:
3928:
3927:Stopping time
3925:
3921:
3918:
3917:
3916:
3913:
3911:
3908:
3906:
3903:
3901:
3898:
3896:
3893:
3891:
3888:
3886:
3883:
3881:
3878:
3876:
3873:
3871:
3868:
3866:
3863:
3861:
3858:
3856:
3853:
3851:
3848:
3846:
3843:
3841:
3838:
3836:
3833:
3831:
3828:
3826:
3823:
3821:
3818:
3816:
3813:
3811:
3808:
3806:
3803:
3801:
3798:
3796:
3793:
3791:
3788:
3786:
3783:
3781:
3778:
3777:
3775:
3771:
3765:
3762:
3760:
3757:
3755:
3752:
3750:
3747:
3745:
3742:
3741:
3739:
3737:
3733:
3726:
3722:
3718:
3717:HewittâSavage
3714:
3710:
3706:
3702:
3701:Zeroâone laws
3699:
3697:
3694:
3692:
3689:
3687:
3684:
3682:
3679:
3677:
3674:
3672:
3669:
3667:
3664:
3662:
3659:
3657:
3654:
3652:
3649:
3648:
3646:
3642:
3636:
3633:
3631:
3628:
3626:
3623:
3621:
3618:
3616:
3613:
3611:
3608:
3606:
3603:
3601:
3598:
3596:
3593:
3591:
3588:
3586:
3583:
3581:
3578:
3576:
3573:
3571:
3568:
3566:
3563:
3562:
3560:
3556:
3550:
3547:
3545:
3542:
3540:
3537:
3535:
3532:
3530:
3527:
3525:
3522:
3521:
3519:
3517:
3513:
3507:
3504:
3502:
3499:
3497:
3494:
3492:
3489:
3488:
3486:
3484:
3480:
3474:
3471:
3469:
3466:
3464:
3461:
3459:
3456:
3454:
3451:
3449:
3446:
3444:
3441:
3439:
3436:
3434:
3431:
3429:
3426:
3424:
3421:
3419:
3416:
3414:
3411:
3409:
3406:
3404:
3401:
3399:
3398:BlackâScholes
3396:
3394:
3391:
3389:
3386:
3384:
3381:
3380:
3378:
3376:
3372:
3366:
3363:
3361:
3358:
3356:
3353:
3351:
3348:
3346:
3343:
3341:
3338:
3337:
3335:
3333:
3329:
3323:
3320:
3318:
3315:
3311:
3308:
3306:
3303:
3302:
3301:
3300:Point process
3298:
3296:
3293:
3291:
3288:
3286:
3283:
3279:
3276:
3274:
3271:
3270:
3269:
3266:
3264:
3261:
3259:
3258:Gibbs measure
3256:
3254:
3251:
3249:
3246:
3245:
3243:
3239:
3233:
3230:
3228:
3225:
3223:
3220:
3218:
3215:
3213:
3210:
3206:
3203:
3201:
3198:
3196:
3193:
3191:
3188:
3187:
3186:
3183:
3181:
3178:
3176:
3173:
3171:
3168:
3166:
3163:
3162:
3160:
3156:
3150:
3147:
3145:
3142:
3140:
3137:
3135:
3132:
3130:
3127:
3125:
3122:
3120:
3117:
3115:
3112:
3110:
3107:
3103:
3100:
3098:
3095:
3094:
3093:
3090:
3088:
3085:
3083:
3080:
3078:
3075:
3073:
3070:
3068:
3065:
3063:
3060:
3058:
3055:
3053:
3050:
3048:
3047:ItĂŽ diffusion
3045:
3043:
3040:
3038:
3035:
3033:
3030:
3028:
3025:
3023:
3022:Gamma process
3020:
3018:
3015:
3013:
3010:
3008:
3005:
3003:
3000:
2998:
2995:
2993:
2990:
2988:
2985:
2983:
2980:
2978:
2975:
2971:
2968:
2966:
2963:
2961:
2958:
2956:
2953:
2951:
2948:
2947:
2946:
2943:
2939:
2936:
2935:
2934:
2931:
2929:
2926:
2924:
2921:
2920:
2918:
2916:
2912:
2904:
2901:
2899:
2896:
2894:
2893:Self-avoiding
2891:
2889:
2886:
2885:
2884:
2881:
2879:
2878:Moran process
2876:
2874:
2871:
2869:
2866:
2864:
2861:
2859:
2856:
2854:
2851:
2849:
2846:
2845:
2843:
2841:
2840:Discrete time
2837:
2833:
2826:
2821:
2819:
2814:
2812:
2807:
2806:
2803:
2796:
2792:
2788:
2785:
2781:
2778:
2774:
2773:
2765:
2761:
2756:
2751:
2746:
2741:
2737:
2733:
2729:
2725:
2721:
2714:
2711:
2705:
2700:
2696:
2692:
2688:
2681:
2678:
2673:
2667:
2663:
2656:
2653:
2648:
2644:
2640:
2636:
2632:
2628:
2623:
2618:
2614:
2610:
2606:
2599:
2596:
2591:
2587:
2582:
2577:
2573:
2569:
2565:
2561:
2557:
2553:
2549:
2542:
2539:
2534:
2530:
2525:
2520:
2515:
2510:
2506:
2502:
2498:
2491:
2488:
2483:
2479:
2474:
2469:
2465:
2461:
2457:
2453:
2449:
2445:
2441:
2434:
2431:
2427:
2422:
2419:
2415:
2409:
2406:
2400:
2397:
2392:
2386:
2382:
2378:
2374:
2367:
2364:
2357:
2353:
2350:
2348:
2345:
2343:
2340:
2338:
2335:
2333:
2330:
2328:
2325:
2323:
2320:
2319:
2315:
2313:
2311:
2306:
2299:
2297:
2294:
2292:
2287:
2280:
2278:
2262:
2258:
2254:
2249:
2245:
2241:
2238:
2216:
2212:
2189:
2185:
2162:
2158:
2154:
2149:
2145:
2141:
2136:
2132:
2128:
2125:
2103:
2099:
2076:
2072:
2049:
2045:
2035:
2019:
1990:
1975:
1971:
1963:
1961:
1954:
1952:
1950:
1940:
1937:
1935:
1932:
1929:
1928:
1924:
1921:
1919:
1916:
1913:
1912:
1908:
1905:
1903:
1900:
1897:
1896:
1892:
1889:
1887:
1884:
1881:
1880:
1876:
1873:
1871:
1868:
1865:
1864:
1860:
1857:
1855:
1852:
1849:
1848:
1844:
1841:
1839:
1836:
1833:
1832:
1828:
1825:
1823:
1820:
1817:
1816:
1812:
1809:
1807:
1804:
1801:
1800:
1796:
1793:
1791:
1788:
1785:
1784:
1780:
1777:
1775:
1772:
1769:
1768:
1765:
1760:
1753:
1746:
1741:
1739:
1733:
1726:
1723: +
1719:
1716: =
1715:
1711:
1704:
1684:
1679:
1669:
1666:
1663:
1659:
1650:
1646:
1642:
1637:
1634:
1631:
1627:
1621:
1617:
1613:
1608:
1604:
1600:
1595:
1591:
1583:
1582:
1581:
1575:
1573:
1572:
1563:
1561:
1559:
1555:
1551:
1544:
1537:
1530:
1526:
1521:
1518:
1516:
1511:
1509:
1504:
1502:
1494:
1490:
1486:
1482:
1478:
1473:
1469:
1455:
1452:
1449:
1446:
1441:
1437:
1431:
1427:
1423:
1420:
1417:
1412:
1408:
1404:
1401:
1396:
1392:
1388:
1385:
1379:
1372:
1369:
1348:
1345:
1342:
1339:
1334:
1330:
1324:
1320:
1316:
1313:
1310:
1305:
1301:
1297:
1294:
1289:
1285:
1281:
1275:
1268:
1265:
1256:
1252:
1249: =
1248:
1244:
1240:
1236:
1232:
1228:
1225: =
1224:
1220:
1217: =
1216:
1196:
1190:
1184:
1181:
1178:
1171:
1170:
1169:
1167:
1163:
1158:
1154:
1134:
1126:
1123:
1120:
1116:
1109:
1106:
1101:
1097:
1089:
1088:
1087:
1069:
1066:
1063:
1058:
1054:
1048:
1044:
1040:
1037:
1032:
1028:
1024:
1019:
1015:
1011:
1005:
999:
992:
991:
990:
987:
983:
979:
975:
955:
952:
949:
944:
934:
931:
928:
924:
915:
911:
907:
902:
892:
889:
886:
882:
873:
869:
865:
860:
857:
854:
850:
844:
840:
836:
831:
827:
823:
818:
814:
806:
805:
804:
800:
796:
792:
788:
784:
780:
775:
771:
752:
749:
746:
743:
738:
734:
730:
725:
721:
717:
712:
708:
704:
701:
694:
693:
692:
690:
683:
679:
674:
670:
663:
656:
649:
645:
640:
638:
634:
614:
608:
605:
600:
596:
578:
566:
565:
564:
562:
559:The ultimate
554:
552:
550:
545:
537:
535:
532:
527:
523:
519:
514:
510:
503:
484:
481:
476:
472:
466:
463:
460:
455:
452:
449:
445:
441:
438:
435:
430:
427:
424:
420:
416:
411:
407:
403:
398:
395:
392:
388:
380:
379:
378:
376:
372:
367:
363:
359:
354:
350:
346:
341:
336:
315:
312:
309:
305:
297:
293:
287:
284:
281:
277:
273:
268:
265:
262:
258:
250:
249:
248:
245:
241:
237:
233:
229:
224:
220:
216:
212:
207:
203:
199:
195:
190:
186:
182:
174:
172:
170:
166:
162:
157:
155:
151:
147:
143:
139:
135:
131:
127:
123:
119:
115:
111:
106:
104:
100:
96:
91:
75:
72:
69:
49:
41:
37:
33:
29:
22:
3985:Econometrics
3947:Wiener space
3835:ItĂŽ integral
3736:Inequalities
3625:Self-similar
3595:GaussâMarkov
3585:Exchangeable
3565:CĂ dlĂ g paths
3501:Risk process
3453:LIBOR market
3322:Random graph
3317:Random field
3164:
3129:Superprocess
3067:LĂ©vy process
3062:Jump process
3037:Hunt process
2873:Markov chain
2852:
2794:
2790:
2727:
2723:
2713:
2694:
2690:
2680:
2661:
2655:
2612:
2608:
2598:
2555:
2551:
2541:
2504:
2500:
2490:
2447:
2443:
2433:
2421:
2413:
2408:
2399:
2372:
2366:
2352:Superprocess
2307:
2303:
2295:
2288:
2284:
2036:
1973:
1969:
1967:
1958:
1948:
1946:
1758:
1751:
1744:
1742:
1737:
1731:
1724:
1717:
1713:
1709:
1702:
1700:
1579:
1567:
1557:
1553:
1549:
1542:
1535:
1528:
1524:
1522:
1519:
1514:
1512:
1507:
1505:
1500:
1498:
1492:
1488:
1484:
1480:
1476:
1254:
1250:
1246:
1242:
1238:
1234:
1230:
1226:
1222:
1218:
1214:
1212:
1165:
1161:
1156:
1152:
1150:
1085:
985:
981:
977:
973:
971:
798:
794:
790:
786:
782:
778:
773:
769:
767:
688:
681:
677:
672:
668:
661:
654:
647:
641:
636:
632:
629:
563:is given by
558:
548:
543:
541:
530:
525:
521:
517:
512:
508:
501:
499:
374:
365:
361:
357:
352:
348:
342:
334:
332:
243:
239:
235:
231:
222:
218:
214:
210:
205:
201:
197:
193:
188:
184:
178:
164:
158:
153:
149:
145:
133:
129:
125:
124:equals
121:
109:
107:
31:
25:
4030:Ruin theory
3968:Disciplines
3840:ItĂŽ's lemma
3615:Predictable
3290:Percolation
3273:Potts model
3268:Ising model
3232:White noise
3190:Differences
3052:ItĂŽ process
2992:Cox process
2888:Loop-erased
2883:Random walk
2327:Random tree
1233:) for
768:Therefore,
345:random walk
200:), and let
4040:Statistics
3820:Filtration
3721:Kolmogorov
3705:Blumenthal
3630:Stationary
3570:Continuous
3558:Properties
3443:HullâWhite
3185:Martingale
3072:Local time
2960:Fractional
2938:pure birth
2622:1503.04558
2358:References
803:) . Thus,
520:, and let
360:, and let
213:in period
3952:Classical
2965:Geometric
2955:Excursion
2615:: 40â49.
2572:2041-210X
2464:1063-5157
2255:−
2242:−
2155:−
2142:−
2129:−
1667:−
1635:−
1453:≥
1450:⋯
1346:≥
1343:⋯
1124:−
1067:⋯
953:⋯
932:−
890:−
858:−
750:≤
747:⋯
744:≤
731:≤
718:≤
585:∞
582:→
482:−
446:∑
436:−
373:over all
278:∑
230:over all
99:genealogy
4088:Category
4073:Category
3957:Abstract
3491:BĂŒhlmann
3097:Compound
2780:Archived
2764:28182672
2724:PLOS ONE
2647:15335040
2639:26626088
2590:29938014
2533:29069434
2482:25575504
2316:See also
1373:″
1269:′
217:, where
118:expected
95:surnames
3580:Ergodic
3468:VaĆĄĂÄek
3310:Poisson
2970:Meander
2755:5300154
2732:Bibcode
2581:5993341
2524:5920349
2473:4395845
1925:0.3304
1909:0.3297
1893:0.3288
1877:0.3276
1861:0.3262
1845:0.3244
1829:0.3221
1813:0.3192
1797:0.3156
660:,
653:,
347:. Let
183:. Let
3920:Tanaka
3605:Mixing
3600:Markov
3473:Wilkie
3438:HoâLee
3433:Heston
3205:Super-
2950:Bridge
2898:Biased
2762:
2752:
2668:
2645:
2637:
2588:
2578:
2570:
2531:
2521:
2480:
2470:
2462:
2387:
1941:0.331
1933:0.3109
1917:0.3051
1901:0.2975
1885:0.2878
1869:0.2751
1853:0.2584
1837:0.2362
1821:0.2058
1151:Since
781:, and
646:. Let
128:where
3773:Tools
3549:M/M/c
3544:M/M/1
3539:M/G/1
3529:Fluid
3195:Local
2643:S2CID
2617:arXiv
1805:0.163
1701:with
500:with
340:= 1.
333:with
3725:LĂ©vy
3524:Bulk
3408:Chen
3200:Sub-
3158:Both
2760:PMID
2666:ISBN
2635:PMID
2586:PMID
2568:ISSN
2529:PMID
2478:PMID
2460:ISSN
2385:ISBN
1221:and
226:are
30:, a
3305:Cox
2750:PMC
2740:doi
2699:doi
2695:110
2627:doi
2613:390
2576:PMC
2560:doi
2519:PMC
2509:doi
2468:PMC
2452:doi
2377:doi
2291:ODE
1789:0.1
575:lim
371:iid
223:n,i
206:n,i
159:In
140:by
97:in
26:In
4090::
3723:,
3719:,
3715:,
3711:,
3707:,
2758:.
2748:.
2738:.
2728:12
2726:.
2722:.
2693:.
2689:.
2641:.
2633:.
2625:.
2611:.
2607:.
2584:.
2574:.
2566:.
2554:.
2550:.
2527:.
2517:.
2505:67
2503:.
2499:.
2476:.
2466:.
2458:.
2448:64
2446:.
2442:.
2383:.
2375:.
2312:.
2034:.
1938:20
1930:10
1922:19
1906:18
1890:17
1874:16
1858:15
1842:14
1826:13
1810:12
1794:11
1740:.
1525:hâČ
1491:=
1479:=
1424:12
1241:=
1164:,
1160:â
989::
801:â1
753:1.
590:Pr
171:.
105:.
3727:)
3703:(
2824:e
2817:t
2810:v
2766:.
2742::
2734::
2707:.
2701::
2674:.
2649:.
2629::
2619::
2592:.
2562::
2556:9
2535:.
2511::
2484:.
2454::
2393:.
2379::
2263:5
2259:p
2250:4
2246:p
2239:1
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2213:p
2190:4
2186:p
2163:3
2159:p
2150:2
2146:p
2137:1
2133:p
2126:1
2104:3
2100:p
2077:2
2073:p
2050:1
2046:p
2020:n
2015:N
1991:i
1986:X
1974:i
1970:n
1949:d
1914:9
1898:8
1882:7
1866:6
1850:5
1834:4
1818:3
1802:2
1786:1
1762:2
1759:p
1755:1
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1748:0
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1738:d
1735:2
1732:p
1728:1
1725:p
1721:0
1718:p
1714:d
1710:d
1706:0
1703:d
1685:.
1680:2
1676:)
1670:1
1664:m
1660:d
1656:(
1651:2
1647:p
1643:+
1638:1
1632:m
1628:d
1622:1
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1614:+
1609:0
1605:p
1601:=
1596:m
1592:d
1558:z
1556:(
1554:h
1550:Ό
1546:3
1543:p
1539:2
1536:p
1532:1
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1508:z
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1495:.
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1483:(
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1402:+
1397:2
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1386:=
1383:)
1380:z
1377:(
1370:h
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1340:+
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1317:3
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1298:2
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1290:1
1286:p
1282:=
1279:)
1276:z
1273:(
1266:h
1255:z
1253:(
1251:h
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1235:z
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1229:(
1227:h
1223:y
1219:z
1215:y
1197:.
1194:)
1191:d
1188:(
1185:h
1182:=
1179:d
1166:d
1162:d
1157:m
1153:d
1135:.
1132:)
1127:1
1121:m
1117:d
1113:(
1110:h
1107:=
1102:m
1098:d
1070:.
1064:+
1059:2
1055:z
1049:2
1045:p
1041:+
1038:z
1033:1
1029:p
1025:+
1020:0
1016:p
1012:=
1009:)
1006:z
1003:(
1000:h
986:i
982:p
978:z
976:(
974:h
956:.
950:+
945:3
941:)
935:1
929:m
925:d
921:(
916:3
912:p
908:+
903:2
899:)
893:1
887:m
883:d
879:(
874:2
870:p
866:+
861:1
855:m
851:d
845:1
841:p
837:+
832:0
828:p
824:=
819:m
815:d
799:m
795:d
791:m
787:j
783:d
779:d
774:m
770:d
739:2
735:d
726:1
722:d
713:0
709:d
705:=
702:0
689:m
685:0
682:d
678:m
673:m
669:d
665:2
662:p
658:1
655:p
651:0
648:p
637:Ό
633:Ό
615:.
612:)
609:0
606:=
601:n
597:Z
593:(
579:n
549:λ
544:1
531:i
526:i
522:X
518:i
513:i
509:S
505:0
502:S
485:i
477:j
473:X
467:1
464:+
461:i
456:1
453:=
450:j
442:=
439:1
431:1
428:+
425:i
421:X
417:+
412:i
408:S
404:=
399:1
396:+
393:i
389:S
375:i
366:i
362:X
358:i
353:i
349:S
338:0
335:Z
316:i
313:,
310:n
306:X
298:n
294:Z
288:1
285:=
282:i
274:=
269:1
266:+
263:n
259:Z
244:n
240:Z
236:i
232:n
219:X
215:n
211:i
202:X
198:n
194:n
189:n
185:Z
165:Ό
154:Ό
150:Ό
146:Ό
134:Ό
130:Ό
126:Ό
122:n
76:1
73:+
70:n
50:n
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
