# 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.

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### Tight bounds for finite de Finetti's theorem

de Finetti's theorem roughly states that infinite sequence of exchangeable random variables are conditionally independent. I am looking for tight bounds for de Finetti's theorem in the following ...
2answers
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### Lower bounds on discrete time finite Markov chains hitting probabilities

I am interested in some general theorems related to lower bounds on discrete time finite Markov chains hitting probabilities (preferably ergodic chains , but not necessarily ), with references . ...
0answers
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### Discrepancy of the finite approximation of the Lebesgue measure

Let $\mu$ be a probabilistic measure on the unit square $Q$ which is the average of $N$ delta-measures in some points in this square; let $\lambda$ denote the Lebesgue measure on $Q$. What is the rate ...
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108 views

### Expectation of maximum of multivariate Gaussian

Given a multivariate Gaussian $\mathbf{X} \sim \mathcal{N}(\mathbf{\mu},\Sigma)$, I believe it is a difficult question to find a closed form for $$\mathbb{E}[ \max\{X_1,\ldots,X_d\}].$$ However, the ...
1answer
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### Comparison of several topologies for probability measures

Let $X$ be a compact metric space and denote $\mathcal M(X)$ the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\operatorname{supp} \mu$ for the support of $\mu$. As is well ...
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### Spectral radius of Markov averaging operator on graphs

The definition of Markov operator which I am familiar with: For a graph $G=(V,E)$, Markov's operator upon a function $\varphi:V\rightarrow\mathbb{C}$ , $\varphi\in L^2(G,\nu)$ ($\nu(v):=\deg(v)$) ...
1answer
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### Supremum of log(E[X]]-E[log(X)]

I've been tackling some questions on probability theory and got stuck on this one. Determine $$\sup_{1≤X≤b} \ \log(E[X])-E[\log(X)]$$ where $X$ is a random variable defined in $[1,b]$. In other words,...
1answer
340 views

### Growing a chain of unit-area triangles: Fills the plane?

Define a process to start with a unit-area equilateral triangle, and at each step glue on another unit-area triangle.                     $50$ ...
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2answers
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### Finite-sample deviation bound of empirical distribution from true distribution

Let $P=(p_1,\ldots,p_k) \in \Delta_k$ be distribution supported on set of size $k$ and let $\hat{P}_n$ be an empirical version of $P$ based on an iid sample of size $n$. Question What's a good non-...
1answer
56 views

### Uniform inequality of the form $\text{Proba}(\sup_{v \in [-M,M]^k}|p^Tv-\hat{p}_n^Tv| \le \epsilon_n) \ge 1 - \delta$

Let $M > 0$, $k$ be a positive integer, and $\mathcal V:=[-M,M]^k$. Finally, let $p \in \Delta_k$, (where $\Delta_k$ is the $(k-1)$-dimensional probability simplex) and let $\hat{p}_n$ be an ...
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### Random Two-Player Asymmetric Game

About half a year ago I asked a question on MSE about a random two player game. At the time, the question received some attention and some progress was made, but was not resolved completely. I have ...
1answer
70 views

### The expectation of partition times needed separate two elements in a set

I met a problem which can be formulated as set partition. Given a set $S=\{s_1,s_2,...,s_n\}$ having $n$ elements, I want to separate two elements, say $s_1,s_2$, in $S$ by repeatedly using set ...
1answer
266 views

### E[X|Y] and E[Y|X]

Suppose $x, y$ are random variables jointly distributed on $[0,1]^2$. The marginal distribution of $x$ is uniform. It is also known that $E[y]=E[x]=\frac12$ and $E[x|y]=y$, so $y$ second-order ...
1answer
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### Information for discovering an item-colour assignment in a combinatorial game

We are given a set $S=\{i_1, i_2, \ldots, i_n\}$ of items and a set $C=\{c_1, c_2, \ldots, c_m\}$ of colours. Each item in $S$ is tinted with one colour $c\in C$. Let $\mathcal{A}$ be the set of all ...
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### Large Deviation of Triple Poisson Product

Let $X_i$ with $i=1,\ldots,n$ be independent Poisson variables, $X_i$ with parameter $\lambda_i.$ Let $\circ$ be a group operation on a group of size $n.$ I would like to obtain a large deviation ...
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### Correlation between square of normal random variables

Suppose I have $X,Y$ bivariate normal with correlation coefficient $\rho \in (0,1)$ . Then , what is the correlation between $X^2$ and $Y^2$ ? I am aware of the fact that the square of the normal ...