# Tagged Questions

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

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### Precise estimate for probability an $n$-point set has diameter smaller than $1$

This question was inspired by an earlier question that I answered but would like a more precise bound for. Consider random points $x_1, \dots, x_n$ in the unit ball in $\mathbb R^d$, uniformly and ...
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### L^1 maximal inequalities for the Ornstein-Uhlenbeck semigroup in infinite dimension

For an infinite-dimensional Gaussian random vector $X$ consider the Ornstein-Uhlenbeck maximal operator: $M f(X) := \sup_{\rho \in [0,1]} \mathsf{E} [f(\rho X + (1-\rho^2)^{1/2} X^\prime) \mid X]$ (...
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### derivation of a gap related to extreme value theory

I have an expression to evaluate as follow: $\mathbb{E}\left[\sum_{k=1}^K s_k f(x_k)\Big|s_k=s_k^{\ast} \right]$ where $\{s_k^\ast\}$ can be treated as a ${policy}$ which is defined as follows: \...
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### On the assumptions in the Berry-Esseen Theorem

Let $X_1,\ldots X_n$ be i.i.d. random variables and denote by $S_n$ their sum. Assuming that $\mathbb{E}S_n=0$, $\mathbb{E}S^2_n=1$ and that $\mathbb{E}|X_i|^3=b$, the Berry-Esseen Theorem (in the i.i....
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### Independence in mathematics

While trying to think about possible interesting notions of algebraic independance over a skew field, I am wondering where in mathematics appears the notion of being independent, or free over ...
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### Are the elementary predictable processes dense in $L^2([M])$ for $M$ a local martingale?

The question is the one from the title. I know this is true when $M$ is an $L^2$ bounded martingale (which is often used in the classical approach to the construction of the stochastic integral) but I'...
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### Eigenvalue perturbation of a symmetric matrix by a random orthogonal projection

Given fixed real symmetric $D\in\mathbb{R}^{n\times n}$ with $n$ distinct eigenvalues, let $U$ be a random orthogonal matrix selected uniformly from the space of $n\times n$ orthogonal matrices, and ...
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### Lower bounding the probability that a zero-mean sequence of random variables stays positive

Assume that $X_n$ is a sequence of a zero-mean and unit variance random variables (and maybe having density w.r.t. to Lebesgue). Can we conclude that $P(X_n \in [0,R_n])$ is bounded away from zero ...
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### Are the coefficients of a linear combination of random vectors as random?

Given are $2n$ random vectors $x_i,y_i\in\mathbb{C}^n$ for $i=1,\ldots,n$ which entries are drawn iid from some absolutely continuous distribution. Every set of $n$ different of those vectors is ...
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### Functional limit theorem under random change of time

FINAL EDIT: There is one main question left: According to the answer, we have choosen $\theta=1$ , where we could choose $0<\theta<\infty$ as we like. His this sufficient, if we regarde the ...
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### An inequality involving conditional variance and its connection to information theory

Given absolutely continuous random variables $(X, Y)$ with joint distribution $P_{XY}$, we construct $Z:=\sqrt{\gamma} Y+N_\mathsf{G}$ where $N_\mathsf{G}\sim N(0, 1)$ and is independent of $(X,Y)$ ...