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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How to approach Basketball “Beat the Pro” drill with Markov Chain

Suppose, similarly to Gambler's ruin problem a Basketball player is doing "Beat the Pro" drill. That is, for every shot made, he scores one point, and for those missed, two points are to be deducted. ...
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Upper bound for randomly weighted sum of independent random variables

I have a sequence of independent random variables {$\epsilon_j$} with mean 0. I also have another sequence of Bernoulli random variables $\delta_1, \delta_2,\dots$ which are dependent on the previous ...
67 views

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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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 ...
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Does the LHS of $E[X_n | \mathscr F_{n-1}]$ make sense even if $X_n$ is not integrable or adapted?

Let $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ be a filtered probability space. Then $X_n$ is a $(\{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)-$martingale if: $X_n$'s are ...
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How can I mathematically prove this time series when $e_t$ has i.i.d distribution?

Not really sure on how to simplify the $y_t$ because there is $y_{t-1}^2$ In order for a time series to be Martingale difference sequence the expected value given all the past value should be 0 and ...
17 views

Is unit root the same with martingale

I'm confused between these two concepts. Could you give an explanation? Many thanks.