# 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.

5,271 questions

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### Expected number of times of choosing a word out of a given vocabulary when words are grouped into overlapping bins

Two players (player C and player G) are playing a (modified) word guessing game. Both players share the same vocabulary $V$ and words in $V$ are grouped into $K$ bins, denoted as $b_1$, $b_2$, ..., $...

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### Evaluate a pair of integrals involving dilogarithms over the unit interval

These are two variations on the "Bonus round" problem, expertly address by student at the end of his answer to A pair of integrals involving square roots and inverse trigonometric functions over the ...

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49 views

### Expected number of times of choosing a word out of a given vocabulary when words are grouped into non-overlapping bins

Two players (player C and player G) are playing a (modified) word guessing game. Both players share the same vocabulary $V$ and words in $V$ are grouped into $K$ bins, denoted as $b_1$, $b_2$, ..., $...

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77 views

### The norm of isotropic sub-Gaussian random vector may not be sub-Gaussian

Suppose $X$ is a isotropic sub-Gaussian $n$-dimensional random vector (i.e. $EXX^T=I_n$, and for any unit vector $u$,$\|\left<X,u\right>\|_{\psi_2}\le K$). It is said that $\|X\|_2-\sqrt n$ may ...

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48 views

### Conditioning on an irrelevant variable in a martingale control problem

Suppose I have two independent Brownian motions $B^1_t, B^2_t$ and $\mathbb F_t$ be the natural filtration generated by them. Let $T > 0$ be a fixed finite number. Let $q_t$ be a $[-1,1]$ valued $\...

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70 views

### Is there a name for “splitting a probability distribution into independent components”?

Suppose I have a random variable $\theta=(\theta_1,\dotsc,\theta_n)$; where the $\theta_i$ might have pairwise correlations. I decompose it into $\theta=\hat\theta(\phi_1,\dotsc,\phi_k)$, where $\hat\...

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83 views

### Prove that a sub-Gaussian random vector over a finite set $S \subset\mathbb R^n$ implies that $|S|$ is exponentially large

Let $X$ be an isotropic random vector (i.e. $E[XX^T]=I_n$) and $X$ takes value in a finite set $S \subset\mathbb R^n$. If $X$ is a sub-Gaussian random vector and the norm $\|X\|_{\psi_2}\le C$ where $...

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70 views

### Counter-example of Subsequence Criterion? [migrated]

The last argument shows that if $X_n\to X_\infty$ a.s. and $N(n)\to\infty$ a.s., then $X_{N(n)}\to X_\infty$.
We have written this out with care because the analogous result for convergence in ...

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55 views

### Expectation of a Random Matrix that Contains Wishart Form

I am interested in calculating the expectation of the following random matrix:
$$A=WX(X^TWX)^{-1},$$
where $W \sim W_p(n,I)$ is a $p\times p$ random Wishart matrix, and $X$ is a fixed $p\times m$ ...

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26 views

### Distribution of a post-selected random variable with power-law distribution

Background
Assume $X \sim \mathcal D$ is a random variable, distributed according to some distribution $\mathcal D$. Then postselection with respect to some set $A$ is defined as the conditional ...

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89 views

### What happens to the Gaussian volume of a Borel set when it is translated?

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb R^n$, $A \subseteq \mathbb R^n$ be Borel and $c \in \mathbb R^n$. Define the translate $A_c := c + A := \{c+a \mid a \in A\} = \{x \in \...

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23 views

### Reference Request: Total Variation Between Dependent and Independent Bernoulli Processes

Let $X$ be a random variable taking values in $\{0,1\}^n$ with the following distribution. For each coordinate $i$, we have $p_i = P(X_i = 1) = c/\sqrt n$, where $c$ is a (very small) constant. ...

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49 views

### Asymptotic rate for the expected value of the square root of sample average

I have iid random variables $X_1, \dots, X_n$ with $X_i \geq 0$, $E[X_i]=1$ and $V[X_i] = \sigma^2$.
Let $S_n = \frac{\sum_{i=1}^n X_i}{n}$.
I'd like to say that $E[\sqrt{S_n}] = 1-O(1/n)$.
My first ...

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49 views

### Order statistics of correlated bivariate Gaussian

Suppose $(X_1,Y_1),...,(X_n,Y_n)$ are i.i.d. bivariate Gaussian with mean zero. Each coordinate has variance 1 and correlation between coordinates is $\rho\in[-1,1]$.
I'm interested in the following ...

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61 views

### Gaussian isoperimetry for $\ell_p$ norms

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb R^n$. It is well-known (e.g see Proposition 1) that for a given Gaussian volume content, half-spaces $H=\{x \in \mathbb R^n | a^Tx \le b\}$ ...