# 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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### 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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### Semi-martingale vs. martingale. What is the difference?

Can someone please explain (preferably in layman's terms) what is the difference between semi-martingales and martingales? I have found the following sentence on Wikipedia: In probability theory, a ...
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### Is it a valid algorithm to win at the casino roulette?

I would like to try the following algorithm in order to win in the roulette: Be an observer until there are 3 same parity numbers in a row ($0$ has no defined parity in this context) Once there were ...
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### Power martingales for change detection: M goes to zero?

I'm trying to apply the power martingale framework by [Vovk et al., 2003] to change detection in unlabeled data streams, just like in [Ho and Wechsler, 2007]. The basic idea involves using a power ...
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### Distribution of $\frac{1}{1+X}$ if $X$ is Lognormal

Suppose $Z \sim \mathcal{N}(0,1)$. Suppose $X$ is a lognormally distributed random variable, defined as $X:=X_0exp^{(-0.5\sigma^2+\sigma Z)}$, in other words, $X$ is log-normal with $\mathbb{E}[X]=X_0$...
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### Carré du champ operator is a quadratic variation

Let $X_t$ be a real valued Markov process (starting at $x$) with generator $L$. Let $\Gamma(f)$ denote Carré du champ operator i.e. $L(f^2) - 2f \cdot L (f)$. As far as I know under suitable ...
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### Are these Bernoulli variables independent?

I was reading a paper in which it was assumed that $\varepsilon_1,\cdots,\varepsilon_n$ conditional on $X$ possess serial (non-linear) dependence, such that P[\varepsilon_t\geq0\mid\...
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### Azuma's inequality Use

For binary vectors, x = (x1,x2,...,xn) and y=(y1,y2,...,yn), the Hammond distance between them is defined by d(x,y) = |x1 - y1| + |x2 - y2| + ... + |xn - yn|. Let A be a finite set of such vectors ...