# Questions tagged [martingale]

In probability theory, a 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.

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### Distribution Function of Standard Normal is a U.I. Martingale?

I'm a little lost on how to show how $X_{t}=\Phi(\frac{W_{t}}{\sqrt{T-t}})$ $0\leq t\leq T$, where $W_{t}$ is the usual Brownian Motion, is a Uniformly Integrable Martingale? My goal is to try and ...
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### Uniformly Integrable Martingale

I have $(Y_{n})_{n\in\mathbb{N}}$ as a seq. of positive, independent r.v.'s whose expectation is 1 $\forall n$. I have the canonical filtration $\mathcal{F}_{n}=\sigma(Y_{k},k\leq n)$. I have already ...
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### How to show that this particular card game is a martingale?

An ordinary deck of cards is randomly shuffled and then the cards are exposed one at a time. At some time before all the cards have been exposed you must say “next”, and if the next card exposed is a ...
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### Conditional expectation of this stochastic process?

I'm just beginning to learn about stochastic processes and encountered this very elementary problem that confused me a bit: We toss a coin that lands on Head with probability $p$ and Tail with $q=1-p$....
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### Show that the manhattan distance from the origin of an ubiased random walk in $\mathbb{Z}^2$ defines a martingale sequence

Consider the infinite lattice $N \times N$. A pebble starting at the origin walks at random, each time moving equiprobably to one of its four neighbors. Let $X_i$ be the distance from the origin, ...
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### martingale convergence theorem

Let $(X_n)_{n\geq 1}$ be i.i.d. random variables with density $f$ with respect to Lebesgue measure on $\mathbb R$, and $f(x)>0$ for all $x\in \mathbb {R}$. $g$ is another density (with respect to ...
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### Are there good examples of martingale processes that are not simple random walks?

Are there non-trivial examples of martingale processes that aren't simple random walks? I'm trying to better understand the difference between martingales and simple random walks. They look pretty ...
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### The magic money tree problem

I thought of this problem in the shower, it was inspired by investment strategies. Let's say there was a magic money tree. Every day, you can offer an amount of money to the money tree and it will ...
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### How to check these sequences generated by i.i.d random variables are martingales?

Let $\{Y_n\}_{n\geq 1}$ be a sequence of independent, identically distributed random variables. $P(Y_i=1)=P(Y_i=-1)=\frac12$ Set $S_0=0$ and $S_n=Y_1+...+Y_n$ if $n\geq 1$ I want to check if the ...
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### log transform fixed PH in Cox model - how?

I have survival data to which I am fitting a Cox model with a continuous predictor. The cumulative martingale residual method (supremum test) of Lin, Wei and Ying suggested that both proportional ...
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### When is the posterior distribution equal to the prior?

So I have heard that if the prior distribution is in the subexponential class, applying Bayes rule does not change the belief. I have been trying to find an example of this but I am unable to do so. I ...
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