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Examples of martingales. An unbiased random walk, in any number of dimensions, is an example of a martingale. For example, consider a 1-dimensional random walk where ...
In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Every martingale is a local martingale; every bounded local martingale is a martingale; in particular, every local martingale that is bounded from below is a supermartingale, and every local martingale that is bounded from above is a submartingale; however, a local ...
In this example, the probability of losing the entire bankroll and being unable to continue the martingale is equal to the probability of 6 consecutive losses: (10/19) 6 = 2.1256%. The probability of winning is equal to 1 minus the probability of losing 6 times: 1 − (10/19) 6 = 97.8744%.
Constructive martingales (Schnorr 1971): A martingale is a function : {,} [,) such that, for all finite strings w, () = ((⌢) + (⌢)) /, where ⌢ is the concatenation of the strings a and b. This is called the "fairness condition": if a martingale is viewed as a betting strategy, then the above condition requires that the bettor plays ...
The condition that the martingale is bounded is essential; for example, an unbiased random walk is a martingale but does not converge. As intuition, there are two reasons why a sequence may fail to converge. It may go off to infinity, or it may oscillate. The boundedness condition prevents the former from happening.
The martingale central limit theorem generalizes this result for random variables to martingales, which are stochastic processes where the change in the value of the process from time t to time t + 1 has expectation zero, even conditioned on previous outcomes.
In probability theory, the martingale representation theorem states that a random variable that is measurable with respect to the filtration generated by a Brownian motion can be written in terms of an Itô integral with respect to this Brownian motion.
Uniform integrability is an extension to the notion of a family of functions being dominated in which is central in dominated convergence.Several textbooks on real analysis and measure theory use the following definition: [1] [2]