# sunday football betting lines

Each way betting has mostly been associated with horse racing. Such betting is possible when the racing track has at least four runners and riders. In such cases, bookmakers usually offer one-half, one-third, one-quarter or one-fifth on the odds on the selection, depending on the number of runners on the field at the time of the event. In terms of football, each way betting is fairly common, particularly in the outright betting markets. A good way to keep up with this is through staying up to date with the latest news!

# Borel cantelli martingale betting

What will actually turn out to be useful is that although the bets are now small, the average time until we hit 1 is actually infinite. So long as at each stage you bet exactly enough that, if you win, you recoup all your losses so far, and one extra pound, this has the same overall effect.

Of course, we need to check that we do eventually win a round, which is not guaranteed if the probability of winning conditional on not having yet won decays sufficiently fast. By taking logs and taking care of the approximations, it can be seen that the divergence or otherwise of determines which way this falls. The first part is more of an aside. In a variety of contexts, whether for testing Large Deviations Principles or calculating expectations by integrating over the tail, it is useful to know good approximations to the tail of various distributions.

In particular, the exact form of the tail of a standard normal distribution is not particularly tractable. The following upper bound is therefore often extremely useful, especially because it is fairly tight, as we will see. Let be a standard normal RV. We are interested in the tail probability. The density function of a normal RV decays very rapidly, as the exponential of a quadratic function of x. This means we might expect that conditional on , with high probability Z is in fact quite close to x.

This concentration of measure property would suggest that the tail probability decays at a rate comparable to the density function itself. In fact, we can show that:. Just by comparing derivatives, we can also show that this bound is fairly tight. In particular:. Now for the second part about CLT. The following question is why I started thinking about various interpretations of CLT in the previous post. Suppose we are trying to prove the Strong Law of Large Numbers for a random variable with 0 mean and unit variance.

Suppose we try to use an argument via Borel-Cantelli :. Now we can use our favourite estimate on the tail of a normal distribution. By Borel-Cantelli, we conclude that with probability 1, eventually. This holds for all , and a symmetric result for the negative case. We therefore obtain the Strong Law of Large Numbers. The question is: was that application of CLT valid? It certainly looks ok, but I claim not. The main problem is that the deviations under discussion fall outside the remit of discussion.

CLT gives a limiting expression for deviations on the scale. If in fact it only becomes accurate for , then it is not relevant for estimating. One solution might be to find some sort of uniform convergence criterion for CLT, ie a hopefully rapidly decreasing function such that. This is possible, as given by the Berry-Esseen theorem , but even the most careful refinements in the special case where the third moment is bounded fail to give better bounds than.

Adding this error term will certainly destroy any hope we had of the sum being finite. Of course, part of the problem is that the supremum in the above definition is certainly not going to be attained at any point under discussion in these post- deviations.

We really want to take a supremum over larger-than-usual deviations if this is to work out. By this stage, however, I hope it is clear what the cautionary note is, even if the argument could potentially be patched. CLT is a theorem about standard deviations.

Separate principles are required to deal with the case of large deviations. In this post, we consider Brownian motion as a Markov process, and consider the recurrence and transience properties in several dimensions. As motivation, observe from Question 5 of this exam paper that it is a very much non-trivial operation to show that Brownian motion in two-dimensions almost surely has zero Lebesgue measure.

We would expect this to be true by default, as we visualise BM as a curve. So it is interesting to see how much we can deduce without significant technical analysis. In this context we assume that for :. This means that almost surely, BM returns to zero infinitely many times. This is easiest shown by using the time-reversal equivalence to deduce that. That is,. These sequence have a nice property that their means remain constant over time this follows directly from the law of iterated expectation.

I thought that this was weird and noteworthy in a why-did-I-never-see-this-before kind of way. Let , and for all , define. Consider the partial sums. This is again a martingale. Applying this to , we have. Now even though the are a martingale sequence we hope that they correspond to a fair game , our total loss can be infinite with probability 1. Note: Since every martingale is a submartingale, we in fact have a submartingale sequence that tends to!

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These sequence have a nice property that their means remain constant over time this follows directly from the law of iterated expectation. I thought that this was weird and noteworthy in a why-did-I-never-see-this-before kind of way. Let , and for all , define. Consider the partial sums. This is again a martingale. Applying this to , we have. Now even though the are a martingale sequence we hope that they correspond to a fair game , our total loss can be infinite with probability 1.

Note: Since every martingale is a submartingale, we in fact have a submartingale sequence that tends to! You are commenting using your WordPress. You are commenting using your Google account. You are commenting using your Twitter account.

You are commenting using your Facebook account. Notify me of new comments via email. Notify me of new posts via email. Also, the probability that we return to zero is the same as the probability that we ever hit 1, since after one time-step they are literally the same problem after symmetry. But if the expected number of visits to anywhere ie the sum across all places is finite, this is clearly ridiculous, since we are running the process for an infinite time, and at each time-step we must be somewhere!

What will actually turn out to be useful is that although the bets are now small, the average time until we hit 1 is actually infinite. So long as at each stage you bet exactly enough that, if you win, you recoup all your losses so far, and one extra pound, this has the same overall effect.

Of course, we need to check that we do eventually win a round, which is not guaranteed if the probability of winning conditional on not having yet won decays sufficiently fast. By taking logs and taking care of the approximations, it can be seen that the divergence or otherwise of determines which way this falls. The first part is more of an aside.

In a variety of contexts, whether for testing Large Deviations Principles or calculating expectations by integrating over the tail, it is useful to know good approximations to the tail of various distributions. In particular, the exact form of the tail of a standard normal distribution is not particularly tractable. The following upper bound is therefore often extremely useful, especially because it is fairly tight, as we will see.

Let be a standard normal RV. We are interested in the tail probability. The density function of a normal RV decays very rapidly, as the exponential of a quadratic function of x. This means we might expect that conditional on , with high probability Z is in fact quite close to x. This concentration of measure property would suggest that the tail probability decays at a rate comparable to the density function itself. In fact, we can show that:. Just by comparing derivatives, we can also show that this bound is fairly tight.

In particular:. Now for the second part about CLT. The following question is why I started thinking about various interpretations of CLT in the previous post. Suppose we are trying to prove the Strong Law of Large Numbers for a random variable with 0 mean and unit variance. Suppose we try to use an argument via Borel-Cantelli :. Now we can use our favourite estimate on the tail of a normal distribution. By Borel-Cantelli, we conclude that with probability 1, eventually.

This holds for all , and a symmetric result for the negative case. We therefore obtain the Strong Law of Large Numbers. The question is: was that application of CLT valid? It certainly looks ok, but I claim not. The main problem is that the deviations under discussion fall outside the remit of discussion.

CLT gives a limiting expression for deviations on the scale. If in fact it only becomes accurate for , then it is not relevant for estimating. One solution might be to find some sort of uniform convergence criterion for CLT, ie a hopefully rapidly decreasing function such that. This is possible, as given by the Berry-Esseen theorem , but even the most careful refinements in the special case where the third moment is bounded fail to give better bounds than.

Adding this error term will certainly destroy any hope we had of the sum being finite. Of course, part of the problem is that the supremum in the above definition is certainly not going to be attained at any point under discussion in these post- deviations. We really want to take a supremum over larger-than-usual deviations if this is to work out.

By this stage, however, I hope it is clear what the cautionary note is, even if the argument could potentially be patched. CLT is a theorem about standard deviations. Separate principles are required to deal with the case of large deviations. In this post, we consider Brownian motion as a Markov process, and consider the recurrence and transience properties in several dimensions.

As motivation, observe from Question 5 of this exam paper that it is a very much non-trivial operation to show that Brownian motion in two-dimensions almost surely has zero Lebesgue measure. We would expect this to be true by default, as we visualise BM as a curve. So it is interesting to see how much we can deduce without significant technical analysis. In this context we assume that for :.

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However, the exponential growth of the bets eventually bankrupts its users due to finite bankrolls. Stopped Brownian motion , which is a martingale process, can be used to model the trajectory of such games. The term "martingale" was introduced later by Ville , who also extended the definition to continuous martingales.

Much of the original development of the theory was done by Joseph Leo Doob among others. Part of the motivation for that work was to show the impossibility of successful betting strategies in games of chance. A basic definition of a discrete-time martingale is a discrete-time stochastic process i. That is, the conditional expected value of the next observation, given all the past observations, is equal to the most recent observation.

Similarly, a continuous-time martingale with respect to the stochastic process X t is a stochastic process Y t such that for all t. It is important to note that the property of being a martingale involves both the filtration and the probability measure with respect to which the expectations are taken.

These definitions reflect a relationship between martingale theory and potential theory , which is the study of harmonic functions. Given a Brownian motion process W t and a harmonic function f , the resulting process f W t is also a martingale. The intuition behind the definition is that at any particular time t , you can look at the sequence so far and tell if it is time to stop.

An example in real life might be the time at which a gambler leaves the gambling table, which might be a function of their previous winnings for example, he might leave only when he goes broke , but he can't choose to go or stay based on the outcome of games that haven't been played yet. That is a weaker condition than the one appearing in the paragraph above, but is strong enough to serve in some of the proofs in which stopping times are used. The concept of a stopped martingale leads to a series of important theorems, including, for example, the optional stopping theorem which states that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial value.