Talk:Martingale (probability theory)

684:

5,000 in a row all the time - but you never do. The way to count the odds is the difference between the frequency of each count in a row. For example, 5 in a row would take X number of flips to see one occurance of 5 in a row. Well it takes a LOT more flips to see a 10 in row than a 5 in a row. The longer the streak, the less often it occurs. Let's just say for example 10 in a row happens once every 100,000 times and and 11 in a row happens once every 110,000 times. The odds are .909 lower (100/110) that 10 in a row will continue to 11 in a row. However, if you took the results of the 11th flip after 1,000 ten-in-a-row runs, it would be 50/50. But we ARE NOT talking about doing 10 in a row many times and then going for the 11 in a row afterwards. We are only talking about 1 series and betting against that 1 series until it reverses. By waiting until there is a higher number of flips in a row (starting at 10 in a row, say), before starting the opposite betting, what you are doing is betting ONLY during a small slice of time, where a larger slice of time is required to get to 11 in a row. You are in effect exploiting the known temporary anomaly in the odds. I've discussed this at length in the past with a oddsmaking math genius - it is correct and could be proven with a qbasic program. The odds of 50/50 flips runs in small peaks and valleys. Imagine a timeline fluctuating up and down as it goes - these are the flips recorded as a fluctuating line. Up is heads, down is tails, left to right are the flips. Up one notch for heads, back to one below the line for tails and vice versa. As long as you get the same in row, the peak or valley gets larger. As soon as the series breaks, you go to position #1 on the other side of the line. The line would look something like seismograph printout. But on our paper, no matter how long our sheet was, you'd almost NEVER find a 10 valley or a 10 peak (10 tails or 10 heads in a row). And it's that much more difficult to get to 11, 12. It's not that you are beating the odds, it's that you are only playing when the odds are in your favor. The longer you let the steak get before you bet, the better the odds it will reverse while you still have money. If you don't believe me, challenge yourself to prove it. Let's write the qbasic code for this proof and post it here. BTW, the reason why no one benefits from this odds anomaly is that you'd have to stand by the roulette wheel all day just to see a goodly enough number in a row that your starting point would be affordable. You might have to wait all week to see one occurance of 10 in a row (must count 0 and 00 as streak continue - so your odd are less than 50/50 to start). So, if you went against it at 10, it would likely reverse by 13. So you bet 100(11-lose), bet 200(12-lose), bet 400(13-win). You lost $ 300, won $ 400, net $ 100 and you worked all week to get it. The anomaly is there, but the risk and time required to exploit it is too great. No matter how many times you double bet before you finally win, in the end, your net winnings equals the amount of your 1st bet.

3240:

distribution is somehow invariant of the history of the system. In fact it's the opposite. The history (i.e., all of the prior realizations) tell you which conditional probability distribution (or at least conditional expected value) to use to describe the next sample. If it is a martingale, then the conditional expected value (which is parameterized by the entire history of the process at that point) will always be equal to the most recent realization. In other words, if I only knew the current observation, my expected value for the next observation might be different if I knew the entire history. Imagine drawing cards out of a deck without replacement. If all you know is the previous card drawn, you have a lot of uncertainty about the next card drawn. However, if you know the entire history (i.e., every card drawn so far), then your expected value of the next card changes substantially. It is that latter expected value that is used in the definition of the martingale. This is why the

780:

theory and the theory of martingales. When you say "given that we tossed a coin 10 times, what's the probability that it will land on heads?" You calculate the conditional probability of an 11th heads given 10 heads, which by Bayes Rule would amount to the equation: P/P = (1/2)^11/(1/2)^10 = 1/2. There you have it, 1/2 probability of getting a heads on the next toss. I'm not trying to discredit you, but I find it highly unlikely that you spoke at length with a "math genius" on this topic. Furthermore, you cannot "prove" such a thing with a computer simulation (lol @ qbasic). Also, given that you are discussing in the mathematics section, please try to use mathematically precise language.

764:$ 20,480? Are you going to go from seeing 100 in a row (which NEVER happens) to 110 in a row (which NEVER NEVER happens). It's not a "fallacy" it's an odds-smoothing anomaly and it could be exploited, but the amount you could bet is too tightly controlled by the demands for a HUGE capital reserve, so even if you waited for a string of many in a row before you start, it wouldn't be worth the effort. To understand this, you need to step back from the absolute of the total odds being spread over all flipping occurrences and see that localized odds-smoothing anomalies do rarely occur and when they do, they are detectable. 3356:

isn't a martingale. Yes, the probability distribution shape may change depending on earlier realisations, but not its center of gravity. For example, a drunkard may lurch with greater amplitudes if it has stumbled into the bar more recently than the coffee shop, but provided the walk stays unbiased it stays a martingale. Likewise, the random idling of a unicyclist can be a martingale despite being likely to go back and forth more than side to side (hence the probability distribution depends on the last two observations, but the expectation value depends only on the last one).

2942:

the mean of that distribution is equivalent to the previous sample does not imply that the process is "equally likely" to "rise" and "fall" unless you put strict qualifications on what it means to rise and fall. You can imagine a probability distribution where 99% of the time it rises but the 1% of the time it falls it falls tremendously. In that case, you still have a Martingale, but the process would spend most of its time rising. So I prefer the old definition perhaps with a little rounding off of edges. This "lay definition" goes too far (IMHO). —

74: 53: 173: 163: 142: 601:

outcomes there is no need to refer to previous results to draw this conclusion. When betting in a "supermartingale" situation, such as a bet on black on a roulette wheel with two green spots, your expectation is determined by the odds of one trial, again without reference to previous trials, and this is largely self-evident without lengthy mathematical proof (I'm not saying we shouldn't show the proof! Only that it's fair to make a plain language statement of the obvious fact as well).

3325:(given all prior observations). In that case, the conditional expectation is 0, notwithstanding the current observation. As they summarize, "on average", the chain's value "does not change, regardless of" what the current value actually is. Again, this definition assumes a Markov chain (explicitly, for simplicity), but a similar form could be used here in the general definition. If a Markov chain is needed to help explain, then it should be stated 2091:

fair coin comes up heads and you lose a dollar every time that fair coin turns up tails. Your winnings are a martingale. That is, your winnings now are the expected value (the "mean" in probability) of your winnings after the next coin flip. In other words, the next coin flip will increase your winnings by a dollar with 50% probability and will decrease your winnings by a dollar with a 50% probability, and so your expended

636:

even more rare. If you withheld the start of your betting until a major statistical anomoly was already present, then what are are actually doing is beting AGAINST the continuation of the MAJOR anomoly. The key would be to know how often 10, 15 and 20, etc. in a row actually occur. Just by the seat of my pants, I'd venture to say you almost never see 20 in a row - even with millions of flips.

22: 2103:. Does that help? Do you think there is something that could be added to the lede that could make things more clear? At the moment, the wikilinks should help clarify some of the muddiness. Where that doesn't help, some of the rest of the page should. However, the lede is a little short, and so maybe something (but what?) could be added. — 3575:

the uncertainty of future outcomes. Thus, the expected value of the next outcome given knowledge of the present and all prior outcomes may be higher than the current outcome if a winning strategy is used. Martingales exclude the possibility of winning strategies based on game history, and thus they are a model of fair games.

3045:

clarification of exactly this point. (The approach is to start simple and move toward more technical as the article proceeds. And not to proceed too long before getting to an example and motivation. The concept of martingales really shouldn't be too difficult for someone unfamiliar with expectation values to understand.)

822:

into play when you're comparing two of the same observations-- a run of 100 heads is unlikely to be followed by another identical run of 100 heads, but reversion to the mean says nothing about observing the next 1 or 10 flips, as that's a different variate. In practice, here is what reversion to the mean means:

3033:"..is a stochastic process (i.e., a sequence of random variables) given all of the previous observed values (i.e., the realizations), the conditional expected value (i.e., the expectation or mean of all cases that share the same previous observed values) for the next value is equal to the current value." 3574:

To contrast, in a process that is not a martingale, it may still be the case that the expected value of the process at one time is equal to the expected value of the process at the next time. However, knowledge of the prior outcomes (e.g., all prior cards drawn from a card deck) may be able to reduce

3059:

Lastly, somewhere in the article I think an example should explain more clearly why the definition is not simply E(X_{n+1}|X_n)=X_n since the obvious examples are actually not conditional on X_1,..X_{n-1}. Similarly, a more tangible example (or at least less abstractly explained) of a martingale with

2055:

I had the same confusion reading the article for the first time. It says that martingale is "a model of a fair game where knowledge of past events never helps predict the mean of the future winnings and only the current event matters." This is just misleading, and sounds like a special case of Markov

821:

216.153.214.89, you're still wrong, and committing the gambler's fallacy by misunderstanding reversion to the mean. Reversion to the mean states that, e.g., a run of 9 heads out of 10 is likely to followed by the next 10 flips having fewer than 9 heads, closer to 5. Reversion to the mean only comes

779:

What are you talking about? To illustrate the point: I just flipped my quarter and it landed heads three times in a row. Is it going to magically be more likely to land on tails when I toss it again? Of course not. If you suggest that it is so, you are contradicting practically all of probability

3448:

Something about the biased coin as an example of a sub- and supermartingale seems misleading. The expectation value of a biased coin doesn't depend on its history – it's always p. Even in the case where you have a coin and are told that it's biased (but not by how much), you wouldn't know whether it

3383:

the history of how you got here, it wouldn't help you to predict where you would go next. Thus, that's conditioned on the entire history and does not require a Markov property. As mentioned in my previous edit, some authors add the Markov property for simplicity, but it is not the general definition

3359:

agr, this suggests a great simplification of the lead. For example, is the definition "a martingale is a sequence of random observations for which the latest realised value is always the expectation value for the following observation" anything short of completely rigorous? Can you fully elucidate a

857:

There is no citation of the etymology of Martingale. In fact, I am reasonably confident that the stochastic process called a Martingale had its origin from the horse collar, not the gambling system. It is to give the imagery that there is a constraint on where the horse's head can move in the next

279:

etc. is quite what we want, but the issue is a little messy since I've been trying to avoid talking about filtrations. The definition of stopping time I'm used to is with respect to a filtration; the natural filtration to use (given that we aren't talking about filtrations) would seem to be the one

3378:

by Motwani and Raghavan. Now scroll down to page 87. It says that the knowledge of the past bets does not help to predict the future. That's the essence of a martingale, and perhaps this is where agr's "notwithstanding" came from. In a Martingale, if you know all of your past history, you still are

829:

2) This applies in say, IQ tests. There are many more people who have an IQ score of 130 than 140. So if you take an IQ test and get a score of 135, it's more likely that you were a person with an IQ score of 130 scoring 5 points high out of luck than a person with a score of 140 scoring 5 points

752:

No, you are wrong. In any lot of odds-smoothing series, there will be outlier events. The longer a continous run is from the mean, the greater the odds that it must revert to the mean. The only fallacy about the "gamblers fallacy" is that some idiots think that all people who look at this and think

683:

No - you are wrong. The odds are added differently than you think. If you flip a coin, it's 50/50. If you've flipped and gotten 10 heads in a row, is it still 50/50 on the 11th? No. Why? Because the more you get of the same in a row, the less likely the streak is to continue. If not, then you'd see

635:

Zadi, the idea of doubling your bet until you win can work, but you'd have to wait until the right point in time. If there was already 10 heads in a row and THEN you started the double bets, your are ARE INDEED much better off. The reason for that is that 10 in a row is very rare and 15 in a row is

3355:

No, you're misimagining it or else it isn't a martingale. If all you know is the current observation, then that is your expectation value for the next observation, and if it's a martingale then you're correct. If knowing more would have changed your expectation to anything other than that, then it

2941:

Hm. I think those edits might go too far. Terms like "likely" and "rise" and "fall" have too much loaded meaning and may communicate the wrong idea to the reader. For example, the conditional probability distribution of the next sample may have significant skew asymmetry. In that case, saying that

2024:

They are only related in that they both describe stochastic processes. The Markov property (roughly) states that after observing the prior state of the system, there is no additional information that can be gained by observing earlier states. In a Markov chain, once we enter one link of the chain,

596:

It seems that more snake oil is being sold, and more ignorance spread, in the form of martingale systems promoted as a sound investment strategy, for stocks or foreign exchange. Shouldn't it perhaps be mentioned that martingale systems have been used, not just by casino gamblers, but by investors

486:

But I am not sure that the remark is placed in the right position (quite at the beginning). On one hand, it is a definition, and should be written in the beginning. On the other hand, it uses some abstract mathematical tools, and should be put at the end of the article in order to avoid to make it

2090:

gives the standard rough definition of a martingale. In a prototypical martingale stochastic process, a realization (i.e., a "draw") of the current value of the stochastic process is exactly the mean of the next value of the process. For example, imagine a game where you win a dollar every time a

600:

On the subject of stating the obvious, perhaps a plain language explanation of the fallacy of martingale systems would be suitable. Simply put, if you you're betting on a flip of a fair coin, then your expectation is zero (that is, your current status +/- 0), and by the very definition of random

3075:

Your qualification of what "always" means suggests a purpose for symbols. In any case, technical jargon can be resolved with Wikilinks as it is done on many other technical pages. I think it's important not to relegate the lead to be an auxiliary example (that is better suited for an "Examples"

3180:

is really the expectation conditioned on having knowledge of all prior realizations. If one of those realizations change, the expectation may change. Moreover, the phrase about martingales being models of fair games is so strong that it seems to suggest that martingales are only applied in the

3018:

Since it is the norm for all mathematical articles to have a separate section for the precise definition, I don't think this article has an exceptional necessity for its lead to be cluttered with a self-referential explanation of that fact. Besides, there's already the TOC. (If there are still

513:

I'm a wikipedia neophyte, so I won't try to sign this or anything like that. Nor am I an expert on sailing vessels, so I won't try to edit this page, either. However-- I do believe there is a type of sail called a martingale, which could benefit from its own page disambiguated from this one.

1985:

I think the term martingale was coined by Ville, somewhat before Lévy used them. About the Girsanov theorem, it only permits to construct a measure that makes SOME Ito processes a martingale (not any Ito process!). And Balsara and Kleinert seem out of place in the references. Why not link to

3580:

This makes no sense. First, it refers to the expected rather than actual value of “the process at one time”. Second, it conflates the outcome of the stochastic process with the outcome of a bet chosen based on skill. And it talks about reducing the “uncertainty” of future outcomes, which is

2925:

I've taken a stab at it myself, philosophy being that anybody should be able to at least understand what the article is about (and those who want details will read further for the rigorous definition). Hence I've put the opening sentence into plain English, avoided defining any symbols, and

3239:

The martingale definition restricts the conditional probability distribution of the next sample. Each possible prefix (i.e., each sequence of prior realizations) maps to a conditional probability distribution of the next sample. The "notwithstanding" sounds like the conditional probability

3375: 621:

makes sense to, unless somehow the conditions of the game itself are changed... or unless you take the rational approach and constrain your bets, not by tying them to a chain of random past events according to an irrational system, but rather by the limits of one's own tolerance for risk.

3275: 3044:

Yes, it is true that the chance of a martingale rising is not always equal to the chance of it falling. But the expecation for how much it will rise does equal the amount it is expected to fall. Yes, this isn't perfectly unambiguous, which is why my proposal immediately follows up with a

830:

low out of bad luck. So your next IQ test is more likely to be closer to the mean of 100. It's the same thing for height; a 6'10" person is more likely to have 6'6" genes and a great environment than 7'2" genes and a poor environment, simply because many more people have 6'6" genes.

753:"not so", would actually try to make $ $ betting on it. Let me ask you a question: If you had just witenessed 100 heads in a row and had $ 900 billion dollars reserve, are you going to tell me you couldn't win the even money bet that the next would be tails - even for say $ 10: --> 825:

1) Getting eight heads out of ten flips can mean that you're flipping a very biased coin. But it's much more likely that you're flipping a fair coin but got an unusual number of heads. Therefore, we expect that the next ten flips will have a number of heads closer to average,

482:

I added a general mathematical definition for martingales that take value in general topological vector spaces. They have quite a lot a applications in mathematical finance and in stochastic partial differential equations, so I think it is an important definition.

3600:

English has been using the word "martingale" since 1580 at the latest. It originally meant a strap that could be added to a horse's harness to limit how far the horse could lift its head; this was used during training as a check against out-of-control behavior.

3041:. Furthermore, in cases (such as this) where the lead is really not speaking that innacurately (ideally just isn't using strict jargon), a disclaimer isn't warranted at all. (Those seeking a completely rigorous definition will read down to that section, still.) 3278:

by Evans and Rosenthal (note that they explicitly assume that the Martingale is a Markov chain "for simplicity", but they also say that "this is not really necessary"). In particular, you can rephrase the definition in terms of the conditional expectation of

3022:

The lead is always supposed to be informal and "without rigor". (At the very least it should be less rigorous than those "mathematically rigorous definitions" it says are "given below".) Now, the lower section contains the mathematical definition (expressed

3516:(e.g., all prior cards drawn from a card deck) may be able to reduce the uncertainty of future outcomes. Thus, the expected value of the next outcome given knowledge of the present and all prior outcomes may be higher than the current outcome if a winning 880:

A coincidence? The gambling system has an obvious connection to the mathematical concept. I find that hard to believe that it would be coincidence. The entire point of the gambling system is a claim that the optional sampling theorem can be violated.

926:

I think that the R-programmed simulation of a Brownian motion is just a sequence of independent variables, whereas it should be the sum of independent random variables. I haven't changed it, because I'd like someone to provide a second opinion. Thanks.

299:

encodes in some subtle way (e.g. by minute perturbations of the lower-order bits) all of the preceding values. So for the moment I am pulling this paragraph out, since unless I am missing something it looks like the last claim is just plain false:

3545:

at a disadvantage in a game of skill. Life is arguably just as much a game of skill as it is of chance, or none of us would bother reading and writing an online encyclopedia. This may be a small difference in the nature of this article, but it's a

3092:

of the previous cards drawn. Moreover, you could implement a "coin" based on such a card deck and thus the conditional probability of "heads" and "tails" would differ if you knew only the previous realization versus knowing all prior realizations.

604:

A similar plain language explanation might be to point out that, "if it makes sense, after six consecutive losses, to bet 2^6 times your original bet, and if results of each bet are independent, then it makes just as much sense to bet that amount

1986:

Bachelier or some other financial applicant instead of referencing Balsara's book "for futures traders". Likewise, treatises of Doob, Meyer or Neveu are actually about martingales, whereas Kleinert's book seems to be about "a lot of other stuff".

1208: 1447: 3492:

concludes stating that their "proof is simple and elegant and is a nice example of the power of martingales; it seems intractible without them." Could we collect together some examples of other insights given by the theory of martingales?

1262: 1321: 2127:

To clarify, I'm describing a "prototypical" Martingale by focusing on only the immediately prior observation. The broader definition of Martingale states that the conditional expectation of the next observation given

2025:

the next transitions out of the chain only depend on that link; they do not depend on how we got to that link. However, in a Martingale (roughly), the conditional expectation of the next observation given this and

1505: 3512:"To contrast, in a process that is not a martingale, it may still be the case that the expected value of the process at one time is equal to the expected value of the process at the next time. However, 1141: 124: 1920: 1626: 1359: 1702: 585:

In optional sampling theorem condition (a) is redundant since you require (b). Second, the link of the word constant to mathematical_constant is irrelevant - it's not the same meaning/intention

331:" need not mean that the occurrence or non-occurrence of that event is completely determined by those values. It can mean rather that, although the occurrence or non-occurrence of the event τ= 1735: 588:

In Optional Sampling Theorem there is a reference to undefined condition (c): "a gambler with a finite lifetime (which gives conditions (a) and (b)) and a house limit on bets (condition (c))"

229: 1588: 2517:

A martingale (at least the conventional definition of one) is a discrete-time stochastic process (because it requires a notion of "immediately before"). A discrete-time stochastic process

1975: 1949: 1040: 3449:

was biased up or down, so still it's not a sub- or supermartingale. And if you were told that it was biased in a particular direction, the initial expectation value probably isn't 0.5.

3157:

An unbiased random walk is an example of a martingale. Martingales are models of fair games because the expectation of winning on the next round always equals the expectation of losing.

2336:. To "observe" these random variables means to "draw" from them according to their probability distributions just as you would draw from a card deck or flip a coin or pull the arm of a 1812: 1099: 3029:

stochastic process (i.e., a sequence of random variables) the conditional expected value of the next observation, given all the past observations, is equal to the last observation.

1654: 1387: 1550: 2572: 2247: 255:

I think it is a good approach. Gambler's fortune introduces the concept better than trying to define it abstractly (only in mathematical terms), without any real world example.--

2029:

prior observations is equal to this observation. So Martingales need not have a Markov property as the next transition can depend on more than the present state of the system. —

2009:? The definitions are quite similar; one speaks of probabilities, the other of expectations. Surely there must be overlap! Yet the current article breaths narry a word of this. 1066: 3323: 2611:. In a continuous-time stochastic process, there is no way to describe the "next" random variable at any given time. However, in a discrete-time stochastic process, at time 527:"Of course in reality the exponential growth of the bets would quickly bankrupt those foolish enough to use the martingale after even a moderately long run of bad luck." 459: 540:

Maybe yhe sentence should be changed (do we need to say that somebody is foolish?). Anyway, since nobody has infinite worth, after a long run of bad luck there are not

2748: 2512: 2425: 2396: 2334: 2099:

on the next flip is identical to your current winnings. Because the expected winnings on the next flip are equal to your known winnings on this flip, the process is a

3088:

replace the card back into the deck. In that case, the conditional probability distribution for the next card draw depends not only on the previous card drawn but on

2713: 2684: 2609: 2544: 2483: 2454: 2367: 2305: 2276: 2219: 1146: 3550:

difference in world-view. I'm not sure how to proceed from here, beyond pointing this out on a talk page, but the anomaly struck me rather hard at the time, sorry.

2823: 2655: 1392: 3379:

clueless about the future outcome. The point of a martingale is not simply that knowing what you have now fixes your expectation of what you have later, it's that

2797: 2777: 2629: 3076:

section). Regarding the conditional expectation and the other realizations, you need more than just the previous realization if the process does not have the

3688: 114: 729:

P.S. The talk page of an article isn't really the place to go into this in depth, but if you want to discuss it more, feel free to hit me at my talk page.

3703: 219: 3213:

I don't understand what you mean here by "If one of those realizations change, the expectation may change." Isn't that what the definition disallows?--

3693: 1217: 195: 1270: 3698: 3467: 3452: 3011: 888: 837: 90: 3644:

The word does not come from the strap for horses, to my knowledge. If I remember correctly, there were other things that were also called a

3683: 3052:

expectation. Better to stick with what terminology is more likely to be familiar, and save more specialised jargon for outside of the lead.

865: 807: 765: 685: 637: 2755:

So martingales are stochastic processes (i.e., ordered lists of random variables) where the conditional expected value of the probability

2177:. That is, we aren't measuring an output of a system and trying to estimate the internal states of that system. Here, "observation" is a " 787: 3031:" Alas, the current version of the lead not only preserves just as much rigor but also defines more terms and introduces more jargon: 2657:. It is not necessary for a discrete-time stochastic process to be a martingale. However, there is a discrete-time stochastic process 1987: 943: 3630:

Actually, for the word origin we should start with the gambling use, as this is not about etymology but mathematics and probability.

3520:

is used. Martingales exclude the possibility of winning strategies based on game history, and thus they are a model of fair games."

2160:

Thanks, that helps a great deal. I was struggling with stuff like "value of an observation" which I think means "value of an observa

186: 147: 3384:

of a martingale. A martingale is a fair game because there's no way to predict the future based on any knowledge of past events. —

2056:

property. It should be changed to "a model in which the expected value of the future state is always equal to the current state."

1458: 81: 58: 295:

I'm not sure how to make the "independent of the future" style definition work---for example, we could have a martingale where X

3430: 3395: 3340: 3259: 3192: 3104: 2953: 2892: 2868: 2836: 2147: 2114: 2040: 495: 2992:) which is likely to rise just as much as it is likely to fall. That is, given all of the previous observed values (i.e., the 1013:

family of random variables. Also, those random variables may take values in a more general space than just the real numbers.

2926:

incorporated mention of a prototypical well-known example, but fundamentally I've tried to preserve all that was being said.

2132:

prior observations (including this one) is equal to this observation. So it's easier to talk about Martingales when they are

1108: 530:

I think this sentence qualifies as gambler's fallacy as it implies a long run of bad luck would affect future bets badly. --

3150:

for the next value is always equal to the most recent value, notwithstanding all of the earlier observed values (i.e., the

742: 720: 671: 403: 1951:

is a real interval (or the whole real axis, or a semiaxis) then a martingale is called a continuous time martingale. If

33: 3648:

in the past. The origin of the word in mathematics is to my knowledge unkown. There is also a French article about it:

1820: 1593: 1326: 3434: 3399: 3344: 3274:

agr: If you'd like to use the "notwithstanding" language, it may be a good idea to take a hint from the definition in

3263: 3196: 3108: 2957: 2896: 2872: 2840: 2151: 2118: 2044: 1662: 3055:

But I'm happy we're no longer introducing symbols in the lead. (I used the word "always" to replace the use of "t".)

1707: 3555: 3245: 3177: 2164:" or "value of the observables". The term "equal to the observation" needs clarification as well, IMO. -- cheers, 3151: 2993: 2854: 2178: 1558: 892: 841: 503: 1954: 1928: 1019: 3606: 3471: 3456: 3218: 3167: 1069: 811: 769: 689: 641: 359: 869: 705:

If you flip a coin, it's 50/50. If you've flipped and gotten 10 heads in a row, is it still 50/50 on the 11th?

1748: 991: 791: 468:

Why on earth is a Martingale called a Martingale? One sentence on the historical reason would be interesting

3661: 3635: 3621: 3374:

Cesiumfrog: You need the whole history because that's the definition of a Martingale. For confirmation, see

1991: 1075: 939: 625:

Maybe this doesn't sound scientific enough, but sometimes common sense should be expressed in a common way.

3498: 3602: 2173:

Hm. I think "observation" is probably a better word than observable. This topic doesn't really relate to

1631: 1364: 3551: 1513: 568:

I think that we had better to put supmartingale, submartingale, semimartingale together in the article.

550: 534: 264: 39: 531: 172: 2549: 2224: 73: 52: 3466:

Never mind – I realized that the bankroll is the random variable, not the outcome of the coin flips.

3365: 3065: 2931: 2906: 2189:(i.e., an ordered list) of random variables that are typically indexed by something related to time. 2165: 2077: 987: 960: 931: 884: 861: 833: 783: 499: 21: 3569:

The post of 20 June 2014 has been largely ignored. Currently the second paragraph in the lead says

3426: 3391: 3336: 3255: 3214: 3188: 3163: 3100: 3084:). Imagine you had a deck of cards and every time you drew a card off the top of the deck, you did 2949: 2888: 2864: 2832: 2143: 2110: 2036: 1049: 260: 3282: 2057: 194:

on Knowledge. If you would like to participate, please visit the project page, where you can join

89:

on Knowledge. If you would like to participate, please visit the project page, where you can join

3657: 3631: 3617: 3143: 3127: 2981: 2973: 2881:

I've edited the lede to try to make a distinction between observed values and random variables. —

2182: 909: 700: 655: 407: 178: 1203:{\displaystyle \scriptstyle \left(\Omega ,\mathbb {P} ,{\mathcal {F}},{\mathcal {F}}_{i}\right)} 858:

time instant. It was a coincidence that this also happened to be a name of a gambling system.

162: 141: 1442:{\displaystyle \scriptstyle {\mathcal {F}}_{i}\subset {\mathcal {F}}_{j}\subset {\mathcal {F}}} 3664: 3639: 3625: 3610: 3590: 3559: 3502: 3494: 3475: 3460: 3438: 3403: 3369: 3348: 3267: 3222: 3200: 3171: 3147: 3112: 3069: 2997: 2961: 2935: 2909: 2900: 2876: 2844: 2168: 2155: 2122: 2080: 2065: 2061: 2048: 2018: 1995: 1211: 964: 947: 935: 913: 896: 873: 845: 815: 795: 773: 746: 738: 724: 716: 693: 675: 667: 645: 629: 572: 557: 507: 472: 416: 3586: 3488: 3005:

An unbiased random walk is an example of a martingale. Martingales are models of fair games.

2337: 1453: 545: 488: 2718: 2488: 2401: 2372: 2310: 251:

Why is the gambler's fortune called out specifically? Isn't it just a type of random walk?

3486:

I would like to suggest adding a section on applications for martingales. The abstract of

3361: 3181:

analysis of games. A wikilink from games to a generic definition may solve this problem. —

3139: 3077: 3061: 2989: 2967:

Well, for future reference of anybody wanting simpler, here was my proposal for the lead:

2927: 2014: 1977:

is the set of natural numbers it is called a discrete time martingale. {xx{Hidden end}xx}

983: 956: 2689: 2660: 2585: 2520: 2459: 2430: 2343: 2281: 2252: 2195: 806:

and after you have done so, please re-consider my comments in light of that information.

2802: 2634: 955:

I agree. As it stands, the Excel formula and the R command/graphic are inconsistent. --

3421: 3386: 3331: 3250: 3183: 3095: 2944: 2883: 2859: 2827: 2575: 2138: 2105: 2031: 1265: 803: 626: 256: 2782: 2762: 2614: 802:

207: Thanks for the feedback. What I am referring to is reversion to the mean. Please

597:

who perceive some kind of scientifically proven advantage to the use of such systems?

3677: 2174: 2133: 2006: 905: 469: 3649: 3533:

simply requires that all parties have equal access to the same body of information,

3081: 1043: 975: 730: 708: 659: 569: 396: 3582: 1010: 979: 191: 3537:. That means that parties that are willfully ignorant (that is, "stupid" ) are 3019:

concerns, perhaps adding "rigorous" to the section title would alleviate them?)

2850: 2715:

tells you the (conditional) expected value of the probability distribution at

2010: 1738: 462: 168: 86: 1257:{\displaystyle \scriptstyle \left(\Omega ,\mathbb {P} ,{\mathcal {F}}\right)} 3027:) and also restates the definition into words (rigorously), saying it is a " 1102: 3010:

and I disagree with the current version. Much of this pertains directly to

3048:

The use of the (common speech) clause "given .." makes redundant the term

3135: 2985: 2186: 1316:{\displaystyle \scriptstyle \{{\mathcal {F}}_{i}\}_{i\in {\mathcal {I}}}} 2579: 592:

Martingale systems in investment, and the value of stating the obvious

413:"1 = E = E = m² - E. We immediately get E = m²+1" Seems to me we get 609:. If it ever makes sense to bet $ 64.00 (or $ 64,000.00), then it 3581:

irrelevant to the issues at hand. So I’m removing this paragraph.

518:

I added it to the disambig at the top --14:18, 8 March 2006 (UTC)

2087: 406:, putting the link here in case anybody wants to link to it ... 292:, which is what I was aiming for in the definition given here. 997:

Why this section was hidden in the article instead of deleted?

15: 544:: the gambler bankrupts and cannot afford further betting. -- 304:

To say that "the occurrence or non-occurrence of the event τ=

2005:

Surely, there must be some theorems relating martingales to

1961: 1935: 1866: 1715: 1612: 1500:{\displaystyle \scriptstyle \{X_{i}\}_{i\in {\mathcal {I}}}} 1489: 1433: 1417: 1400: 1345: 1305: 1281: 1243: 1183: 1172: 1026: 2136:, but the generic Martingale need not have this property. — 3000:

for the next value is always equal to the current value.

2853:

in physics too. "Observation" really is a better term ("

1136:{\displaystyle \scriptstyle \left(\cdot ,\cdot \right)} 978:

redirects here, Wouldn't it be better if redirected to

3276:

Probability and statistics: the science of uncertainty

1958: 1932: 1711: 1666: 1635: 1597: 1562: 1462: 1396: 1368: 1330: 1274: 1221: 1150: 1112: 1079: 1053: 1023: 494:

Good points. For now I moved it down. See the article

3285: 2805: 2785: 2765: 2721: 2692: 2663: 2637: 2617: 2588: 2552: 2523: 2491: 2462: 2433: 2404: 2375: 2346: 2313: 2284: 2255: 2227: 2198: 1957: 1931: 1823: 1751: 1710: 1665: 1634: 1596: 1561: 1516: 1461: 1395: 1367: 1329: 1273: 1220: 1149: 1111: 1078: 1052: 1022: 419: 3489:

The least variable phase type distribution is Erlang

2369:

may tell you nothing about what you would draw from

190:, a collaborative effort to improve the coverage of 85:, a collaborative effort to improve the coverage of 3616:Please find a good RS reference and we can add it. 2905:Thanks for your efforts. It all helps! -- cheers, 3317: 2817: 2791: 2771: 2742: 2707: 2678: 2649: 2623: 2603: 2566: 2538: 2506: 2477: 2448: 2419: 2390: 2361: 2328: 2299: 2270: 2241: 2213: 2192:For example, a continuous-time stochastic process 1969: 1943: 1915:{\displaystyle \mathbb {E} =\left(X_{i},u\right).} 1914: 1806: 1729: 1696: 1648: 1621:{\displaystyle \scriptstyle i,j\in {\mathcal {I}}} 1620: 1582: 1544: 1499: 1441: 1381: 1354:{\displaystyle \scriptstyle i,j\in {\mathcal {I}}} 1353: 1315: 1256: 1202: 1135: 1093: 1060: 1034: 453: 3060:respect to another sequence could be beneficial. 1697:{\displaystyle \scriptstyle \left(X_{i},u\right)} 3535:if they have the presence of mind to seek it out 1656:, the three following properties are satisfied: 2076:Please explain in English, someone! -- cheers, 1730:{\displaystyle \scriptstyle {\mathcal {F}}_{i}} 3656:, 169, Printemps 2005, DOI: 10.4000/msh.2945) 2485:might allow you to predict a future draw from 2307:, and at time 3.4, there is a random variable 275:I don't think saying that is independent of X 3176:"Notwithstanding" is not the right word. The 2686:with the property that observing ("drawing") 8: 3037:An article should never open with the words 2278:. So, at time 0, there is a random variable 1583:{\displaystyle \scriptstyle u\in V^{\star }} 1477: 1463: 1293: 1275: 3541:at a disadvantage in a game of chance, but 2095:on the next flip is $ 0, and your expected 1970:{\displaystyle \scriptstyle {\mathcal {I}}} 1944:{\displaystyle \scriptstyle {\mathcal {I}}} 1035:{\displaystyle \scriptstyle {\mathcal {I}}} 3652:(Roger Mansuy, “Histoire de martingales”, 487:too hostile for non-mathematical readers. 136: 47: 3650:"Histoire de martingales" by Roger Mansuy 3309: 3290: 3284: 2804: 2784: 2779:given all draws of variables before time 2764: 2720: 2691: 2662: 2636: 2616: 2587: 2560: 2559: 2551: 2522: 2490: 2461: 2432: 2403: 2374: 2345: 2312: 2283: 2254: 2235: 2234: 2226: 2197: 1960: 1959: 1956: 1934: 1933: 1930: 1892: 1871: 1865: 1864: 1858: 1841: 1825: 1824: 1822: 1787: 1772: 1760: 1753: 1752: 1750: 1720: 1714: 1713: 1709: 1676: 1664: 1633: 1611: 1610: 1595: 1573: 1560: 1521: 1515: 1488: 1487: 1480: 1470: 1460: 1432: 1431: 1422: 1416: 1415: 1405: 1399: 1398: 1394: 1366: 1344: 1343: 1328: 1304: 1303: 1296: 1286: 1280: 1279: 1272: 1242: 1241: 1234: 1233: 1219: 1188: 1182: 1181: 1171: 1170: 1163: 1162: 1148: 1110: 1084: 1077: 1051: 1025: 1024: 1021: 439: 418: 1807:{\displaystyle \mathbb {E} <\infty .} 1009:One can define a martingale which is an 1094:{\displaystyle \scriptstyle V^{\star }} 617:make sense to bet that amount, then it 358:, that occurrence or non-occurrence is 138: 49: 19: 2631:the "next" random variable is at time 1323:with the following property: for each 335:may be correllated with the values of 3523:A martingale illustrates a fair game 2582:, if you like)) to a random variable 707:" To which my answer is, yes, it is. 7: 2799:is equal to the value drawn at time 2750:, then that process is a martingale. 1649:{\displaystyle \scriptstyle i\leq j} 1555:is called a martingale if for each 1382:{\displaystyle \scriptstyle i\leq j} 498:and its history for my explanation. 184:This article is within the scope of 79:This article is within the scope of 3689:High-importance Statistics articles 1545:{\displaystyle X_{i}:\Omega \to V,} 284:. In this case is a function of X 38:It is of interest to the following 3654:Mathématiques et sciences humaines 1798: 1530: 1227: 1156: 613:makes sense to do so. If it ever 14: 3704:Mid-priority mathematics articles 3596:False statement about word origin 2567:{\displaystyle k\in \mathbb {N} } 2242:{\displaystyle t\in \mathbb {R} } 204:Knowledge:WikiProject Mathematics 207:Template:WikiProject Mathematics 171: 161: 140: 99:Knowledge:WikiProject Statistics 72: 51: 20: 3694:WikiProject Statistics articles 3565:Reason for removal of paragraph 3514:knowledge of the prior outcomes 224:This article has been rated as 119:This article has been rated as 102:Template:WikiProject Statistics 3244:modifier is so important (see 3119:I would suggest the following: 2849:These comments still hold for 2737: 2725: 2702: 2696: 2673: 2667: 2598: 2592: 2533: 2527: 2501: 2495: 2472: 2466: 2443: 2437: 2414: 2408: 2385: 2379: 2356: 2350: 2323: 2317: 2294: 2288: 2265: 2259: 2208: 2202: 1877: 1859: 1829: 1792: 1788: 1784: 1765: 1761: 1757: 1533: 1214:, that is a probability space 1061:{\displaystyle \scriptstyle V} 774:11:28, 19 September 2008 (UTC) 747:06:05, 18 September 2008 (UTC) 725:06:04, 18 September 2008 (UTC) 694:05:50, 18 September 2008 (UTC) 676:00:20, 13 September 2008 (UTC) 646:00:09, 13 September 2008 (UTC) 429: 423: 308:depends only on the values of 1: 3591:01:01, 17 November 2017 (UTC) 3503:16:50, 12 November 2013 (UTC) 3439:21:01, 1 September 2011 (UTC) 3404:20:21, 1 September 2011 (UTC) 3370:03:44, 1 September 2011 (UTC) 3349:20:07, 1 September 2011 (UTC) 3329:as a device for simplicity. — 3318:{\displaystyle X_{n+1}-X_{n}} 3268:03:08, 1 September 2011 (UTC) 2857:" probably is even better). — 2019:03:40, 20 November 2010 (UTC) 2001:Relation to Markov chains ??? 1143:this duality). Moreover, let 816:01:09, 23 February 2009 (UTC) 796:18:07, 11 February 2009 (UTC) 573:16:29, 9 September 2006 (UTC) 360:probabilistically independent 198:and see a list of open tasks. 93:and see a list of open tasks. 3699:B-Class mathematics articles 2427:is strongly correlated with 2066:18:56, 20 January 2017 (UTC) 970: 508:01:51, 4 December 2005 (UTC) 473:18:52, 22 October 2005 (UTC) 3684:B-Class Statistics articles 3223:21:19, 31 August 2011 (UTC) 3201:18:29, 31 August 2011 (UTC) 3172:12:13, 31 August 2011 (UTC) 3113:18:29, 31 August 2011 (UTC) 3070:02:46, 31 August 2011 (UTC) 2962:19:05, 30 August 2011 (UTC) 2936:01:12, 30 August 2011 (UTC) 2910:20:35, 17 August 2011 (UTC) 2901:17:53, 17 August 2011 (UTC) 2877:17:49, 17 August 2011 (UTC) 2845:14:28, 15 August 2011 (UTC) 2340:. Making an observation of 2169:06:05, 15 August 2011 (UTC) 2156:02:33, 15 August 2011 (UTC) 2123:02:07, 15 August 2011 (UTC) 2081:12:56, 14 August 2011 (UTC) 2049:02:15, 15 August 2011 (UTC) 922:Demonstration of R software 630:04:56, 22 August 2007 (UTC) 3720: 3360:counterexample TedPavlic? 3246:conditional expected value 3178:conditional expected value 3025:E(X_{n+1}|X_1,...,X_n)=X_n 2221:maps each continuous time 2181:" of a random variable. A 1996:13:50, 28 April 2010 (UTC) 1264:equipped with a family of 965:22:18, 14 March 2010 (UTC) 897:21:35, 31 March 2009 (UTC) 874:11:37, 24 April 2008 (UTC) 846:21:31, 31 March 2009 (UTC) 3560:13:37, 20 June 2014 (UTC) 3461:19:56, 29 June 2013 (UTC) 3419:Please review new lede. — 1005:A more general definition 992:23:20, 23 June 2010 (UTC) 914:13:31, 2 April 2009 (UTC) 465:00:49, 21 Aug 2004 (UTC) 454:{\displaystyle E=m^{2}-1} 410:22:40, 14 Jun 2004 (UTC) 402:Just added an article on 399:03:06, 13 Nov 2003 (UTC) 265:16:31, 28 July 2024 (UTC) 223: 156: 118: 67: 46: 3476:03:44, 2 July 2013 (UTC) 3039:"Very roughly speaking," 2546:maps each discrete time 2514:with absolute certainty. 2398:. On the other hand, if 1070:topological vector space 948:16:46, 27 May 2009 (UTC) 699:Yep, that's exactly the 558:13:38, 21 May 2006 (UTC) 535:21:00, 20 May 2006 (UTC) 230:project's priority scale 3665:20:59, 8 May 2024 (UTC) 3640:12:27, 8 May 2024 (UTC) 3626:06:06, 8 May 2024 (UTC) 3611:22:45, 6 May 2024 (UTC) 522: 246: 187:WikiProject Mathematics 3319: 2819: 2793: 2773: 2744: 2743:{\displaystyle M(k+1)} 2709: 2680: 2651: 2625: 2605: 2568: 2540: 2508: 2507:{\displaystyle C(3.4)} 2479: 2450: 2421: 2420:{\displaystyle C(3.4)} 2392: 2391:{\displaystyle C(3.4)} 2363: 2330: 2329:{\displaystyle C(3.4)} 2301: 2272: 2243: 2215: 1971: 1945: 1916: 1808: 1731: 1698: 1650: 1622: 1584: 1546: 1501: 1443: 1383: 1355: 1317: 1258: 1204: 1137: 1095: 1062: 1036: 455: 82:WikiProject Statistics 28:This article is rated 3376:Randomized algorithms 3320: 2820: 2794: 2774: 2745: 2710: 2681: 2652: 2626: 2606: 2569: 2541: 2509: 2480: 2451: 2422: 2393: 2364: 2331: 2302: 2273: 2249:to a random variable 2244: 2216: 1972: 1946: 1917: 1809: 1732: 1699: 1651: 1623: 1585: 1547: 1502: 1444: 1384: 1356: 1318: 1259: 1205: 1138: 1096: 1063: 1037: 1001:{xx{Hidden begin}xx} 456: 3526:of chance, not skill 3381:even if you did know 3283: 2803: 2783: 2763: 2719: 2708:{\displaystyle M(k)} 2690: 2679:{\displaystyle M(k)} 2661: 2635: 2615: 2604:{\displaystyle D(k)} 2586: 2550: 2539:{\displaystyle D(k)} 2521: 2489: 2478:{\displaystyle C(0)} 2460: 2456:, then drawing from 2449:{\displaystyle C(0)} 2431: 2402: 2373: 2362:{\displaystyle C(0)} 2344: 2311: 2300:{\displaystyle C(0)} 2282: 2271:{\displaystyle C(t)} 2253: 2225: 2214:{\displaystyle C(t)} 2196: 1955: 1929: 1925:If the directed set 1821: 1749: 1708: 1663: 1632: 1594: 1559: 1514: 1459: 1393: 1365: 1327: 1271: 1218: 1147: 1109: 1076: 1050: 1020: 971:Polya's urn redirect 902:You may wish to see 417: 210:mathematics articles 2818:{\displaystyle k-1} 2650:{\displaystyle k+1} 105:Statistics articles 3315: 3144:stochastic process 3128:probability theory 2982:stochastic process 2974:probability theory 2815: 2789: 2769: 2740: 2705: 2676: 2647: 2621: 2601: 2564: 2536: 2504: 2475: 2446: 2417: 2388: 2359: 2326: 2297: 2268: 2239: 2211: 2183:stochastic process 1967: 1966: 1941: 1940: 1912: 1804: 1727: 1726: 1694: 1693: 1646: 1645: 1618: 1617: 1580: 1579: 1542: 1497: 1496: 1439: 1438: 1379: 1378: 1351: 1350: 1313: 1312: 1254: 1253: 1200: 1199: 1133: 1132: 1091: 1090: 1058: 1057: 1032: 1031: 853:Origin of the name 564:Compared to others 451: 280:generated by the X 179:Mathematics portal 34:content assessment 3529:. A fair game of 3508:Chance, Not Skill 2792:{\displaystyle k} 2772:{\displaystyle k} 2624:{\displaystyle k} 2072:Lead is gibberish 1212:probability space 951: 934:comment added by 887:comment added by 876: 864:comment added by 836:comment added by 786:comment added by 734: 712: 701:gambler's fallacy 663: 656:Gambler's fallacy 556: 523:Gambler's Fallacy 362:of the values of 244: 243: 240: 239: 236: 235: 135: 134: 131: 130: 3711: 3552:TheLastWordSword 3324: 3322: 3321: 3316: 3314: 3313: 3301: 3300: 3140:random variables 2990:random variables 2907:Michael C. Price 2824: 2822: 2821: 2816: 2798: 2796: 2795: 2790: 2778: 2776: 2775: 2770: 2749: 2747: 2746: 2741: 2714: 2712: 2711: 2706: 2685: 2683: 2682: 2677: 2656: 2654: 2653: 2648: 2630: 2628: 2627: 2622: 2610: 2608: 2607: 2602: 2573: 2571: 2570: 2565: 2563: 2545: 2543: 2542: 2537: 2513: 2511: 2510: 2505: 2484: 2482: 2481: 2476: 2455: 2453: 2452: 2447: 2426: 2424: 2423: 2418: 2397: 2395: 2394: 2389: 2368: 2366: 2365: 2360: 2338:one-armed bandit 2335: 2333: 2332: 2327: 2306: 2304: 2303: 2298: 2277: 2275: 2274: 2269: 2248: 2246: 2245: 2240: 2238: 2220: 2218: 2217: 2212: 2166:Michael C. Price 2078:Michael C. Price 1976: 1974: 1973: 1968: 1965: 1964: 1950: 1948: 1947: 1942: 1939: 1938: 1921: 1919: 1918: 1913: 1908: 1904: 1897: 1896: 1876: 1875: 1870: 1869: 1862: 1857: 1853: 1846: 1845: 1828: 1813: 1811: 1810: 1805: 1791: 1777: 1776: 1764: 1756: 1736: 1734: 1733: 1728: 1725: 1724: 1719: 1718: 1703: 1701: 1700: 1695: 1692: 1688: 1681: 1680: 1655: 1653: 1652: 1647: 1627: 1625: 1624: 1619: 1616: 1615: 1589: 1587: 1586: 1581: 1578: 1577: 1551: 1549: 1548: 1543: 1526: 1525: 1506: 1504: 1503: 1498: 1495: 1494: 1493: 1492: 1475: 1474: 1454:random variables 1448: 1446: 1445: 1440: 1437: 1436: 1427: 1426: 1421: 1420: 1410: 1409: 1404: 1403: 1388: 1386: 1385: 1380: 1360: 1358: 1357: 1352: 1349: 1348: 1322: 1320: 1319: 1314: 1311: 1310: 1309: 1308: 1291: 1290: 1285: 1284: 1263: 1261: 1260: 1255: 1252: 1248: 1247: 1246: 1237: 1209: 1207: 1206: 1201: 1198: 1194: 1193: 1192: 1187: 1186: 1176: 1175: 1166: 1142: 1140: 1139: 1134: 1131: 1127: 1103:topological dual 1100: 1098: 1097: 1092: 1089: 1088: 1067: 1065: 1064: 1059: 1041: 1039: 1038: 1033: 1030: 1029: 950: 928: 899: 859: 848: 798: 732: 710: 661: 553: 548: 460: 458: 457: 452: 444: 443: 247:gamble's fortune 212: 211: 208: 205: 202: 181: 176: 175: 165: 158: 157: 152: 144: 137: 125:importance scale 107: 106: 103: 100: 97: 76: 69: 68: 63: 55: 48: 31: 25: 24: 16: 3719: 3718: 3714: 3713: 3712: 3710: 3709: 3708: 3674: 3673: 3598: 3567: 3510: 3484: 3468:206.124.141.187 3453:206.124.141.187 3446: 3305: 3286: 3281: 3280: 3078:Markov property 2801: 2800: 2781: 2780: 2761: 2760: 2717: 2716: 2688: 2687: 2659: 2658: 2633: 2632: 2613: 2612: 2584: 2583: 2548: 2547: 2519: 2518: 2487: 2486: 2458: 2457: 2429: 2428: 2400: 2399: 2371: 2370: 2342: 2341: 2309: 2308: 2280: 2279: 2251: 2250: 2223: 2222: 2194: 2193: 2074: 2003: 1983: 1981:Coined by Ville 1953: 1952: 1927: 1926: 1888: 1887: 1883: 1863: 1837: 1836: 1832: 1819: 1818: 1768: 1747: 1746: 1712: 1706: 1705: 1672: 1671: 1667: 1661: 1660: 1630: 1629: 1592: 1591: 1569: 1557: 1556: 1517: 1512: 1511: 1476: 1466: 1457: 1456: 1414: 1397: 1391: 1390: 1363: 1362: 1325: 1324: 1292: 1278: 1269: 1268: 1226: 1222: 1216: 1215: 1180: 1155: 1151: 1145: 1144: 1117: 1113: 1107: 1106: 1080: 1074: 1073: 1048: 1047: 1018: 1017: 1007: 999: 973: 929: 924: 889:132.228.195.207 882: 855: 838:132.228.195.207 831: 781: 594: 583: 566: 551: 525: 500:Oleg Alexandrov 480: 478:Recent addition 435: 415: 414: 391: 381: 371: 357: 348: 341: 330: 321: 314: 298: 291: 287: 283: 278: 273: 249: 209: 206: 203: 200: 199: 177: 170: 150: 121:High-importance 104: 101: 98: 95: 94: 62:High‑importance 61: 32:on Knowledge's 29: 12: 11: 5: 3717: 3715: 3707: 3706: 3701: 3696: 3691: 3686: 3676: 3675: 3672: 3671: 3670: 3669: 3668: 3667: 3603:TooManyFingers 3597: 3594: 3578: 3577: 3566: 3563: 3509: 3506: 3483: 3480: 3479: 3478: 3445: 3442: 3417: 3416: 3415: 3414: 3413: 3412: 3411: 3410: 3409: 3408: 3407: 3406: 3357: 3353: 3352: 3351: 3312: 3308: 3304: 3299: 3296: 3293: 3289: 3230: 3229: 3228: 3227: 3226: 3225: 3206: 3205: 3204: 3203: 3160: 3159: 3158: 3155: 3121: 3120: 3116: 3115: 3057: 3056: 3053: 3046: 3042: 3035: 3020: 3008: 3007: 3002: 2965: 2964: 2923: 2922: 2921: 2920: 2919: 2918: 2917: 2916: 2915: 2914: 2913: 2912: 2814: 2811: 2808: 2788: 2768: 2753: 2752: 2751: 2739: 2736: 2733: 2730: 2727: 2724: 2704: 2701: 2698: 2695: 2675: 2672: 2669: 2666: 2646: 2643: 2640: 2620: 2600: 2597: 2594: 2591: 2576:natural number 2562: 2558: 2555: 2535: 2532: 2529: 2526: 2515: 2503: 2500: 2497: 2494: 2474: 2471: 2468: 2465: 2445: 2442: 2439: 2436: 2416: 2413: 2410: 2407: 2387: 2384: 2381: 2378: 2358: 2355: 2352: 2349: 2325: 2322: 2319: 2316: 2296: 2293: 2290: 2287: 2267: 2264: 2261: 2258: 2237: 2233: 2230: 2210: 2207: 2204: 2201: 2125: 2097:total winnings 2073: 2070: 2069: 2068: 2052: 2051: 2002: 1999: 1982: 1979: 1963: 1937: 1923: 1922: 1911: 1907: 1903: 1900: 1895: 1891: 1886: 1882: 1879: 1874: 1868: 1861: 1856: 1852: 1849: 1844: 1840: 1835: 1831: 1827: 1815: 1814: 1803: 1800: 1797: 1794: 1790: 1786: 1783: 1780: 1775: 1771: 1767: 1763: 1759: 1755: 1743: 1742: 1723: 1717: 1691: 1687: 1684: 1679: 1675: 1670: 1644: 1641: 1638: 1614: 1609: 1606: 1603: 1600: 1576: 1572: 1568: 1565: 1553: 1552: 1541: 1538: 1535: 1532: 1529: 1524: 1520: 1491: 1486: 1483: 1479: 1473: 1469: 1465: 1435: 1430: 1425: 1419: 1413: 1408: 1402: 1377: 1374: 1371: 1347: 1342: 1339: 1336: 1333: 1307: 1302: 1299: 1295: 1289: 1283: 1277: 1266:sigma-algebras 1251: 1245: 1240: 1236: 1232: 1229: 1225: 1210:be a filtered 1197: 1191: 1185: 1179: 1174: 1169: 1165: 1161: 1158: 1154: 1130: 1126: 1123: 1120: 1116: 1087: 1083: 1056: 1028: 1006: 1003: 998: 995: 972: 969: 968: 967: 923: 920: 919: 918: 917: 916: 866:124.171.59.106 854: 851: 850: 849: 827: 823: 808:216.153.214.89 800: 799: 766:216.153.214.89 750: 749: 727: 703:. You wrote, " 686:216.153.214.89 681: 680: 679: 678: 649: 648: 638:216.153.214.89 593: 590: 582: 579: 577: 565: 562: 561: 560: 524: 521: 520: 519: 511: 510: 479: 476: 450: 447: 442: 438: 434: 431: 428: 425: 422: 394: 393: 386: 376: 366: 353: 346: 339: 326: 319: 312: 296: 289: 285: 281: 276: 272: 269: 268: 267: 248: 245: 242: 241: 238: 237: 234: 233: 222: 216: 215: 213: 196:the discussion 183: 182: 166: 154: 153: 145: 133: 132: 129: 128: 117: 111: 110: 108: 91:the discussion 77: 65: 64: 56: 44: 43: 37: 26: 13: 10: 9: 6: 4: 3: 2: 3716: 3705: 3702: 3700: 3697: 3695: 3692: 3690: 3687: 3685: 3682: 3681: 3679: 3666: 3663: 3659: 3658:Tensorproduct 3655: 3651: 3647: 3643: 3642: 3641: 3637: 3633: 3632:Limit-theorem 3629: 3628: 3627: 3623: 3619: 3618:Limit-theorem 3615: 3614: 3613: 3612: 3608: 3604: 3595: 3593: 3592: 3588: 3584: 3576: 3572: 3571: 3570: 3564: 3562: 3561: 3557: 3553: 3549: 3544: 3540: 3536: 3532: 3528: 3527: 3521: 3519: 3515: 3507: 3505: 3504: 3500: 3496: 3491: 3490: 3481: 3477: 3473: 3469: 3465: 3464: 3463: 3462: 3458: 3454: 3450: 3443: 3441: 3440: 3436: 3432: 3428: 3424: 3423: 3405: 3401: 3397: 3393: 3389: 3388: 3382: 3377: 3373: 3372: 3371: 3367: 3363: 3358: 3354: 3350: 3346: 3342: 3338: 3334: 3333: 3328: 3310: 3306: 3302: 3297: 3294: 3291: 3287: 3277: 3273: 3272: 3271: 3270: 3269: 3265: 3261: 3257: 3253: 3252: 3247: 3243: 3238: 3237: 3236: 3235: 3234: 3233: 3232: 3231: 3224: 3220: 3216: 3212: 3211: 3210: 3209: 3208: 3207: 3202: 3198: 3194: 3190: 3186: 3185: 3179: 3175: 3174: 3173: 3169: 3165: 3161: 3156: 3153: 3149: 3145: 3141: 3137: 3133: 3129: 3125: 3124: 3123: 3122: 3118: 3117: 3114: 3110: 3106: 3102: 3098: 3097: 3091: 3087: 3083: 3079: 3074: 3073: 3072: 3071: 3067: 3063: 3054: 3051: 3047: 3043: 3040: 3036: 3034: 3030: 3026: 3021: 3017: 3016: 3015: 3013: 3006: 3003: 3001: 2999: 2995: 2991: 2987: 2983: 2979: 2975: 2970: 2969: 2968: 2963: 2959: 2955: 2951: 2947: 2946: 2940: 2939: 2938: 2937: 2933: 2929: 2911: 2908: 2904: 2903: 2902: 2898: 2894: 2890: 2886: 2885: 2880: 2879: 2878: 2874: 2870: 2866: 2862: 2861: 2856: 2852: 2848: 2847: 2846: 2842: 2838: 2834: 2830: 2829: 2812: 2809: 2806: 2786: 2766: 2758: 2754: 2734: 2731: 2728: 2722: 2699: 2693: 2670: 2664: 2644: 2641: 2638: 2618: 2595: 2589: 2581: 2577: 2556: 2553: 2530: 2524: 2516: 2498: 2492: 2469: 2463: 2440: 2434: 2411: 2405: 2382: 2376: 2353: 2347: 2339: 2320: 2314: 2291: 2285: 2262: 2256: 2231: 2228: 2205: 2199: 2191: 2190: 2188: 2184: 2180: 2176: 2175:observability 2172: 2171: 2170: 2167: 2163: 2159: 2158: 2157: 2153: 2149: 2145: 2141: 2140: 2135: 2134:Markov chains 2131: 2126: 2124: 2120: 2116: 2112: 2108: 2107: 2102: 2098: 2094: 2089: 2085: 2084: 2083: 2082: 2079: 2071: 2067: 2063: 2059: 2054: 2053: 2050: 2046: 2042: 2038: 2034: 2033: 2028: 2023: 2022: 2021: 2020: 2016: 2012: 2008: 2007:Markov chains 2000: 1998: 1997: 1993: 1989: 1980: 1978: 1909: 1905: 1901: 1898: 1893: 1889: 1884: 1880: 1872: 1854: 1850: 1847: 1842: 1838: 1833: 1817: 1816: 1801: 1795: 1781: 1778: 1773: 1769: 1745: 1744: 1740: 1721: 1689: 1685: 1682: 1677: 1673: 1668: 1659: 1658: 1657: 1642: 1639: 1636: 1607: 1604: 1601: 1598: 1574: 1570: 1566: 1563: 1539: 1536: 1527: 1522: 1518: 1510: 1509: 1508: 1484: 1481: 1471: 1467: 1455: 1450: 1428: 1423: 1411: 1406: 1375: 1372: 1369: 1340: 1337: 1334: 1331: 1300: 1297: 1287: 1267: 1249: 1238: 1230: 1223: 1213: 1195: 1189: 1177: 1167: 1159: 1152: 1128: 1124: 1121: 1118: 1114: 1104: 1085: 1081: 1071: 1054: 1045: 1014: 1012: 1004: 1002: 996: 994: 993: 989: 985: 981: 977: 966: 962: 958: 954: 953: 952: 949: 945: 941: 937: 933: 921: 915: 911: 907: 903: 901: 900: 898: 894: 890: 886: 879: 878: 877: 875: 871: 867: 863: 852: 847: 843: 839: 835: 828: 824: 820: 819: 818: 817: 813: 809: 805: 797: 793: 789: 788:207.237.81.59 785: 778: 777: 776: 775: 771: 767: 748: 744: 740: 736: 728: 726: 722: 718: 714: 706: 702: 698: 697: 696: 695: 691: 687: 677: 673: 669: 665: 657: 653: 652: 651: 650: 647: 643: 639: 634: 633: 632: 631: 628: 623: 620: 616: 612: 608: 602: 598: 591: 589: 586: 580: 578: 575: 574: 571: 563: 559: 554: 547: 543: 539: 538: 537: 536: 533: 528: 517: 516: 515: 509: 505: 501: 497: 493: 492: 491: 490: 484: 477: 475: 474: 471: 466: 464: 448: 445: 440: 436: 432: 426: 420: 411: 409: 408:Bryan Barnard 405: 400: 398: 389: 385: 379: 375: 369: 365: 361: 356: 352: 345: 338: 334: 329: 325: 318: 311: 307: 303: 302: 301: 293: 270: 266: 262: 258: 254: 253: 252: 231: 227: 221: 218: 217: 214: 197: 193: 189: 188: 180: 174: 169: 167: 164: 160: 159: 155: 149: 146: 143: 139: 126: 122: 116: 113: 112: 109: 92: 88: 84: 83: 78: 75: 71: 70: 66: 60: 57: 54: 50: 45: 41: 35: 27: 23: 18: 17: 3653: 3645: 3599: 3579: 3573: 3568: 3547: 3542: 3538: 3534: 3530: 3525: 3524: 3522: 3517: 3513: 3511: 3495:Gareth Jones 3487: 3485: 3482:Applications 3451: 3447: 3444:Biased Coin? 3420: 3418: 3385: 3380: 3330: 3326: 3249: 3241: 3182: 3152:realizations 3146:) where the 3131: 3094: 3089: 3085: 3082:Markov chain 3058: 3049: 3038: 3032: 3028: 3024: 3009: 3004: 2994:realizations 2977: 2971: 2966: 2943: 2924: 2882: 2858: 2826: 2757:distribution 2756: 2161: 2137: 2129: 2104: 2100: 2096: 2092: 2075: 2030: 2026: 2004: 1988:90.27.21.180 1984: 1924: 1554: 1452:A family of 1451: 1044:directed set 1015: 1008: 1000: 974: 936:Philip Maton 925: 856: 801: 751: 704: 682: 624: 618: 614: 610: 606: 603: 599: 595: 587: 584: 576: 567: 541: 532:Kurulananfok 529: 526: 512: 485: 481: 467: 461:. Typo? - 412: 401: 395: 387: 383: 377: 373: 367: 363: 354: 350: 343: 336: 332: 327: 323: 316: 309: 305: 294: 274: 250: 226:Mid-priority 225: 185: 151:Mid‑priority 120: 80: 40:WikiProjects 3242:conditional 3148:expectation 3050:conditional 2998:expectation 2855:realization 2851:observables 2179:realization 1105:(denote by 1011:uncountable 980:Urn problem 976:Polya's urn 930:—Preceding 883:—Preceding 860:—Preceding 832:—Preceding 782:—Preceding 763:10,240: --> 581:suggestions 546:gala.martin 542:future bets 489:gala.martin 404:filtrations 201:Mathematics 192:mathematics 148:Mathematics 3678:Categories 3646:martingale 3362:Cesiumfrog 3327:explicitly 3132:martingale 3080:(also see 3062:Cesiumfrog 2978:martingale 2928:Cesiumfrog 2101:martingale 1739:measurable 1389:, one has 1068:be a real 984:Squidonius 957:Chadhoward 762:5,120: --> 761:2,560: --> 760:1,280: --> 496:martingale 96:Statistics 87:statistics 59:Statistics 3548:ginormous 3422:TedPavlic 3387:TedPavlic 3332:TedPavlic 3251:TedPavlic 3184:TedPavlic 3096:TedPavlic 2984:(i.e., a 2945:TedPavlic 2884:TedPavlic 2860:TedPavlic 2828:TedPavlic 2574:(i.e., a 2139:TedPavlic 2106:TedPavlic 2032:TedPavlic 804:read this 627:zadignose 607:right now 288:through X 257:FeralOink 3518:strategy 3142:(i.e. a 3136:sequence 2986:sequence 2759:at time 2187:sequence 944:contribs 932:unsigned 906:Melcombe 885:unsigned 862:unsigned 834:unsigned 784:unsigned 759:640: --> 758:320: --> 757:160: --> 470:Torfason 3431:contrib 3396:contrib 3341:contrib 3260:contrib 3193:contrib 3105:contrib 3012:the MOS 2996:), the 2954:contrib 2893:contrib 2869:contrib 2837:contrib 2580:integer 2578:(or an 2148:contrib 2115:contrib 2058:Sprlzrd 2041:contrib 756:80: --> 755:40: --> 754:20: --> 615:doesn't 570:Jackzhp 397:Populus 392:, ... . 349:, ..., 322:, ..., 228:on the 123:on the 30:B-class 3583:Loraof 1361:with 1072:, and 611:always 36:scale. 3531:skill 3134:is a 2980:is a 2185:is a 2011:linas 1628:with 1042:be a 826:five. 733:RETOG 711:RETOG 662:RETOG 654:Eek! 619:never 552:what? 463:Gauge 3662:talk 3636:talk 3622:talk 3607:talk 3587:talk 3556:talk 3499:talk 3472:talk 3457:talk 3427:talk 3392:talk 3366:talk 3337:talk 3256:talk 3248:). — 3219:talk 3189:talk 3168:talk 3130:, a 3101:talk 3066:talk 2976:, a 2950:talk 2932:talk 2889:talk 2865:talk 2833:talk 2144:talk 2111:talk 2093:gain 2088:lede 2086:The 2062:talk 2037:talk 2015:talk 1992:talk 1796:< 1590:and 1101:its 1016:Let 988:talk 982:? -- 961:talk 940:talk 910:talk 893:talk 870:talk 842:talk 812:talk 792:talk 770:talk 690:talk 642:talk 504:talk 261:talk 115:High 3543:are 3539:not 3215:agr 3164:agr 3138:of 3126:In 3090:all 3086:not 2988:of 2972:In 2825:. — 2499:3.4 2412:3.4 2383:3.4 2321:3.4 2162:ble 2130:all 2027:all 1704:is 297:t+1 277:t+1 271:tau 220:Mid 3680:: 3660:-- 3638:) 3624:) 3609:) 3589:) 3558:) 3501:) 3474:) 3459:) 3437:) 3402:) 3368:) 3347:) 3303:− 3266:) 3221:) 3199:) 3170:) 3162:-- 3154:). 3111:) 3068:) 3014:: 2960:) 2934:) 2899:) 2875:) 2843:) 2810:− 2557:∈ 2232:∈ 2154:) 2121:) 2064:) 2047:) 2017:) 1994:) 1799:∞ 1640:≤ 1608:∈ 1575:⋆ 1567:∈ 1534:→ 1531:Ω 1507:: 1485:∈ 1449:. 1429:⊂ 1412:⊂ 1373:≤ 1341:∈ 1301:∈ 1228:Ω 1157:Ω 1125:⋅ 1119:⋅ 1086:⋆ 1046:, 990:) 963:) 946:) 942:• 912:) 904:] 895:) 872:) 844:) 814:) 794:) 772:) 745:) 723:) 692:) 674:) 658:! 644:) 506:) 446:− 427:τ 390:+3 382:, 380:+2 372:, 370:+1 342:, 315:, 263:) 3634:( 3620:( 3605:( 3585:( 3554:( 3497:( 3470:( 3455:( 3435:@ 3433:/ 3429:/ 3425:( 3400:@ 3398:/ 3394:/ 3390:( 3364:( 3345:@ 3343:/ 3339:/ 3335:( 3311:n 3307:X 3298:1 3295:+ 3292:n 3288:X 3264:@ 3262:/ 3258:/ 3254:( 3217:( 3197:@ 3195:/ 3191:/ 3187:( 3166:( 3109:@ 3107:/ 3103:/ 3099:( 3093:— 3064:( 2958:@ 2956:/ 2952:/ 2948:( 2930:( 2897:@ 2895:/ 2891:/ 2887:( 2873:@ 2871:/ 2867:/ 2863:( 2841:@ 2839:/ 2835:/ 2831:( 2813:1 2807:k 2787:k 2767:k 2738:) 2735:1 2732:+ 2729:k 2726:( 2723:M 2703:) 2700:k 2697:( 2694:M 2674:) 2671:k 2668:( 2665:M 2645:1 2642:+ 2639:k 2619:k 2599:) 2596:k 2593:( 2590:D 2561:N 2554:k 2534:) 2531:k 2528:( 2525:D 2502:) 2496:( 2493:C 2473:) 2470:0 2467:( 2464:C 2444:) 2441:0 2438:( 2435:C 2415:) 2409:( 2406:C 2386:) 2380:( 2377:C 2357:) 2354:0 2351:( 2348:C 2324:) 2318:( 2315:C 2295:) 2292:0 2289:( 2286:C 2266:) 2263:t 2260:( 2257:C 2236:R 2229:t 2209:) 2206:t 2203:( 2200:C 2152:@ 2150:/ 2146:/ 2142:( 2119:@ 2117:/ 2113:/ 2109:( 2060:( 2045:@ 2043:/ 2039:/ 2035:( 2013:( 1990:( 1962:I 1936:I 1910:. 1906:) 1902:u 1899:, 1894:i 1890:X 1885:( 1881:= 1878:] 1873:i 1867:F 1860:| 1855:) 1851:u 1848:, 1843:j 1839:X 1834:( 1830:[ 1826:E 1802:. 1793:] 1789:| 1785:) 1782:u 1779:, 1774:i 1770:X 1766:( 1762:| 1758:[ 1754:E 1741:. 1737:- 1722:i 1716:F 1690:) 1686:u 1683:, 1678:i 1674:X 1669:( 1643:j 1637:i 1613:I 1605:j 1602:, 1599:i 1571:V 1564:u 1540:, 1537:V 1528:: 1523:i 1519:X 1490:I 1482:i 1478:} 1472:i 1468:X 1464:{ 1434:F 1424:j 1418:F 1407:i 1401:F 1376:j 1370:i 1346:I 1338:j 1335:, 1332:i 1306:I 1298:i 1294:} 1288:i 1282:F 1276:{ 1250:) 1244:F 1239:, 1235:P 1231:, 1224:( 1196:) 1190:i 1184:F 1178:, 1173:F 1168:, 1164:P 1160:, 1153:( 1129:) 1122:, 1115:( 1082:V 1055:V 1027:I 986:( 959:( 938:( 908:( 891:( 868:( 840:( 810:( 790:( 768:( 743:c 741:/ 739:t 737:( 735:8 731:C 721:c 719:/ 717:t 715:( 713:8 709:C 688:( 672:c 670:/ 668:t 666:( 664:8 660:C 640:( 555:) 549:( 502:( 449:1 441:2 437:m 433:= 430:] 424:[ 421:E 388:t 384:X 378:t 374:X 368:t 364:X 355:t 351:X 347:2 344:X 340:1 337:X 333:t 328:t 324:X 320:2 317:X 313:1 310:X 306:t 290:t 286:1 282:i 259:( 232:. 127:. 42::

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Index