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# Questions tagged [pr.probability]

Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

5,382 questions
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Suppose I have an SDE of the form $$dX_t=b(X_t)dt+\sigma (X_t)dB_t+\int_{\mathbb{R}}G_{t-}(y)N(dtdy)$$ which I can solve weakly if I cut off the last integral to range over the set $\{\mid{y}\mid > ... 1answer 46 views ### Reference request: When is the variance in the central limit theorem for Markov chains positive? I'm looking for a reference which gives sufficient conditions for the variance to be positive in the central limit theorem for Markov chains (cf https://en.wikipedia.org/wiki/... 1answer 61 views ### Maximal correlation and independence Let$X$and$Y$be random variables. Then the maximal correlation$\rho_m(X;Y)$is defined as $$\rho_m (X;Y) := \max_{(f(X),g(Y))\in S} \mathbb{E} [f(X)g(Y)]$$ where$S$is the collection of pairs ... 0answers 51 views ### For a martingale$f_0,f_1,\ldots $how can we bound$P(\frac{1}{n} \|f_n\|) \le 1$for all$ n \ge N)$? Suppose$f_0,f_1, \ldots$is a martingale (or i.i.d sequence) in$\mathbb R^d$with$f_0=0$and all$\|f_n - f_{n-1}\| \le L$say. There are many concentration results for the initial segment of the ... 1answer 95 views ### Behavior of random positive-definite matrix in high dimension? Consider a random matrix$A \in \mathbb{R}^{n\times n}$with i.i.d. entries, with symmetric law and finite variance. I am curious about the behavior of$$\mathrm{Tr}( (A^T A + \lambda \mathrm{Id})^{-1}... 1answer 90 views ### Isoperimetry on$[0, 1]^n$w.r.t$\ell_p$distance, with$p \in [1,\infty]$Let$A$be a measurable subset of the metric space$\mathcal X = ([0, 1]^n,\ell_p)$with$1 \le p \le \infty$, and define its$\varepsilon$-blowup by$A^\varepsilon:=\{x \in \mathcal X \mid \|x-a\|_p \...
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### Good upper-bound for $\mathbb E[|X-np|^r]$ where $X \sim \text{Binomial}(n,p)$ and $r \ge 1$

Disclaimer. Question moved from SE. Setup Let $X \sim \text{Binomial}(p, n)$, and $r \ge 1$. Question What is a good upper-bound for $\mathbb E[|X-np|^r]$ ? Solution for small $r$ If $r=2$, then ...
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### An application of Girsanov's Theorem

Let $(W,H,i)$ be the classical Wiener space where $W=C_0([0,1])$, $H$ is the Cameron-Martin space. Let $A= I_{W}+a$ such that $A:W \rightarrow W$ and $a \in L^{0}(\mu,H)$, $a$ has adapted derivative, ...
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### Do i.i.d Sums Concentrate Any Faster Than Martingales?

Suppose $X_1,X_2, \ldots, X_N \in \mathbb R^d$ are random variables with each $\|X_n\|_2 \le 1/2$ (this choice of the constant simplifies later formulae). The simplest concentration inequality I ...
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### A comprehensive list of random walk inequalities?

I am interested in finding a comprehensive list of all noticeable random walk inequalities. ie. $S_n = \sum_{k\leq n} X_i$ for i.i.d symmetric $X_i$ I can only seem to find books/papers that list ...
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### Maxima of Brownian motion

It is well-known that Brownian motion attains infinitely many maxima in each time interval $[0,T]$ a.s.. From a physics perspective it seems reasonable that when the disorder of the path of a ...
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### Joint drunkard walks

The drunkard walk is a game where two players have $a$ and $b$ dollars, respectively, and they play a series of fair games (both risking one dollar in each game) until one of them goes broke. My ...
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### When does volume ratio of concentric balls decay faster than Gaussian?

Let $\mu$ be a finite measure on $\mathbb{R}^n$. Let $B_1$ to be the unit Euclidean ball centered at 0 in $\mathbb{R}^n$. Therefore, for any $t>0$ and $\theta\in\mathbb{R}^n$, $tB_1+\theta$ is the ...
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In my study, I wish to get the mathematical expectation for the term below. The vector $\boldsymbol{z} \in \mathcal{C}^{N\times1}$ and $\boldsymbol z \sim \mathcal{CN}\left(\boldsymbol{0},\boldsymbol{... 1answer 68 views ### Obtaining a lower bound on the expectation using the Sudakov-Fernique inequality In my work I wish to obtain a lower bound for the term below. Here the expectation is taken over$h$, a standard random Gaussian vector of length$n$. The minimum is taken over all$\{i_1,\dots,i_L\} \...

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