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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420 views

An expansion from Ramanujan related to birthday problem

A friend designed a drinking game with a lucky wheel of 30 distinct icons. When playing, each one takes turn to spin the wheel, and write down the items until the first one who gets the item that has ...
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51 views

Lipschitz function of independent subgaussian random variables

This question was asked here, but I have reason to believe that it's a serious research question appropriate for this forum (also, the answers given at the link aren't satisfactory). ​If $X\in\mathbb{...
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A closed form of mean-field equations

Assume that a system at time t, for example number of costumers in a line at time $t$ which is denoted by $q(t)$, follows a Markov chain with these dynamics (probabilities) $$P(q(t+\Delta t)-q(t)=1)=\...
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101 views

Normal multivariate orthant probabilities

(Previously I posted a similar question on math.SE, hoping that this question would have an easy answer. As the question appears hard, I am hoping I can perhaps get more feedback here.) Let $\mathbf{...
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What can be said about percolation clusters after deleting a positive fraction of edges in general?

Start with a bond-percolation process just above criticality, say $p=1/2+\varepsilon$ on the graph $\mathbb Z^2$ with $\varepsilon>0$. Sample $D\in\{0,1\}^E$ from an independent product measure ...
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71 views

Computing the support of the equilibrium measure in Johansson's 1999 paper “Shape fluctuations and random matrices” in detail?

I am trying to compute the equilibrium measure for the Meixner ensemble on page 19 (on the arxiv version). The "details" of the computation are in Section 6, where he finds the equilibrium measure is ...
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1answer
68 views

Sum of Square of the Eigenvalues of Wishart Matrix

Let $A\in\mathbb{R}^{m\times d}$ matrix with iid standard normal entries, and $m\geqslant d$, and define $S=A^T A$. I want to have a tight upper bound for $\sum_{k=1}^d \lambda_k^2$, where $\...
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127 views

Minimising an Integrated Relative Entropy Functional

Suppose I am given A probability distribution on $\mathbf R^d$, with density $\pi (x)$. A family of transition kernels $\{ q^0 (x \to \cdot) \}_{x \in \mathbf R^d}$ on $\mathbf R^d$, with densities $...
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212 views

Reference request: norm topology vs. probabilist's weak topology on measures

Let $(X,d)$ be a metric space and $\mathcal{M}(X)$ be the space of regular (e.g. Radon) measures on $X$. There are two standard topologies on $\mathcal{M}(X)$: The (probabilist's) weak topology and ...
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45 views

What is the probability that $X_i$ is the $k^{th}$ order statistic in consecutive trials?

Consider $n$ r.vs ${X_1, X_2,...,X_n}$. Each is i.i.d drawn from some distribution $f(.)$. What is the probability that $X_i$ is the $k^{th}$ order statistic in any two consecutive trials?
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540 views

Large-n limit of the distribution of the normalized sum of Cauchy random variables

What is the large-n limit of a distribution of the following sample statistic:$$x\equiv\displaystyle\frac{\sum^{n}X_{i}}{\,\sqrt{\,\sum^{n}X_{i}^{2}\,}\,}$$ when sampling the Cauchy(0,1) distribution? ...
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83 views

A lower bound on the expected sum of Bernoulli random variables given a constraint on its distribution

Given a set of Bernoulli random variables $x_1, \dots, x_n$ (not necessarily identical) with $X= \sum_{0<i\leq n} x_i$, I am intrested in finding a lower-bound for $\frac{\mathbb{E} [ \min (X,k) ]...
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90 views

Hashed coupon collector

The story: A sport card store manager has $r$ customers, that together wish to assemble a $n$-cards collection. Every day, a random customer arrives and buys his favorite card (that is, each customer ...
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73 views

Arithmetic structure of non-zero cumulants

It is known that any non-Gaussian distribution must have infinitely many non-zero cumulants (Marcinkiewicz). I was wondering if something stronger is known about the structure of non-zero cumulants. ...
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65 views

Distribution of a linear pure-birth process' integral

I stumbled across the following random variable, defined as the integral of a linear pure-birth process i.e. a Yule process: $$ Z_t = \mathbb{E}\bigg[\int_0^t Y_s ds \bigg| Y_t=k , Y_0=1\bigg] $$ ...
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1answer
115 views

Symbol for monotone relationship between two probability distributions

Motivation: At the present time it really isn't clear to me why this question might be inappropriate for the MathOverflow. However, it appears that some people are down-voting this question even if ...
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75 views

Probability of a random collection of subsets being a cover

Consider the set $[n]=\{1,2,\ldots,n\}$. Suppose for each set $A\subseteq [n]$ I have a $p_A \in [0,1]$. I now create a random collection $\mathcal{W}\subseteq\mathcal{P}([n])$ of subsets of $[n]$ by ...
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74 views

Average number of pieces of a random piecewise-linear function

Let $I$ be a (nonempty) compact interval in $\mathbb R$ and $a_1,b_1,\ldots,a_L,b_L \in \mathbb R$. Let $\varphi$ be a piecewise function with $T \ge 2$ pieces(for example $T=2$ for the choice $\...
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1answer
141 views

Understanding Finite Size Scaling in Percolation Theory

Fundamental results in percolation theory are all based on the assumption that the system sizes are infinite, as the spanning/percolating cluster is by definition an infinitely sized cluster that ...
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89 views

Linear Independence of random binary vectors

Suppose we have $Y_1, \ldots, Y_n \in \mathbb{R}^m$, $n$ independent random vectors ($m \geq n$), where the entries of each $Y_i$ are i.i.d. Bernoulli random variables taking the values $\{0, 1\}$ ...
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218 views

Slowest initial state for convergence of finite birth-and-death Markov chains

Consider the continuous-time birth-and-death Markov chain on $\{1,\cdots,n\}$ with all rates equal to $1$. Is it true that the convergence to equilibrium, in total variation distance, is slowest when ...
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3answers
144 views

Reference request: book on stochastic calculus (not finance)

I am looking at fractional Gaussian/Brownian noise from a signal theoretic and engineering point of view. In particular, I am looking at the math behind what defines these noise processes and what ...
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249 views

Concerning Luzin-(N)-property

Definition: a function $f:\mathbb{R}\to \mathbb{R}$ has Luzin-(N)-Property if $f$ maps any null set to a null set. By https://www.encyclopediaofmath.org/index.php/Luzin-N-property, it is known that ...
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1answer
133 views

Ratio of integrals with increasing dimension over Euclidean balls

Let $f_n(x)\geq0$ be any sequence of nonnegative $L^1(\mathbb{R}^n)$ functions such that $\int_{\mathbb{R}^{n}}f_n(x)dx=1$ where $dx$ is the Lebesgue measure on $\mathbb{R}^n$. For any $a>1,\...
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1answer
123 views

Lower bound of the expectation of the product of inner products of random vectors

I encountered the following value in my research: Let $n,m$ be some integer. Suppose $\alpha_1,\dots,\alpha_m$ are unit vectors in $\mathbb{R}^n$. Denote $$ L = \mathop{\mathrm{E}}_x[ \prod_{1\...
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1answer
91 views

The distribution of the area of a region cut out by chordal SLE?

Let $\mathbb{D}$ be the unit disc. Let $a,b \in \partial \mathbb{D}$. Let $\gamma$ be a chordal $SLE_{k}$ from $a$ to $b$. For $k \leq 4$, $\gamma$ is a simple curve, and so $\mathbb{D} \setminus \...
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54 views

Optimal tail bounds for the sample mean

Given a bounded random variable $X$ such that $|X| < M$, I want to know how many iid samples $x_1, \dots, x_n \sim X$ have to be drawn such that the sample mean $\bar{x} = (x_1 + \dots + x_n)/n$ ...
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1answer
96 views

The square modulus of coordinates of a uniformly chosen point in complex projective space is uniform in the simplex

I can't recall where I learned this (beautiful) fact, and I would like a reference (if possible, in a textbook): Let $(z_0:\cdots:z_n) \in \mathbb{P}^n(\mathbb{C})$ be chosen uniformly at random w....
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55 views

Bridging between Rosethal Inequalities and log convex tails

Let $X_1,\ldots,X_n$ be independent with $\mathbf{E}[X_i] = 0$ and $\mathbf{E}[|X_i|^t] < \infty$ for some $t \ge 2$. Write $\|X\|_p = (E|X|^p)^{1/p}$. Then we have the classical "Rosenthal-type ...
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1answer
259 views

Sub-Gaussian decay of convolution of $L^1$ function with Gaussian kernel

I think it might be helpful to put the new statement at the beginning and put the original post at the end. This new statement is more mathematically elegant. Let $f\geq0$ be in $L^1(\mathbb{R}^d)$ ...
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60 views

Defining weak solutions to infinitely many SDEs on the same probability space

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 > ...
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On statistics of perfect matchings between planar $4$ colorable and planar $3$ colorable

Does the mean for number of perfect matchings of random graphs that are planar and $3$ colorable much higher than graphs that are planar are not $3$ colorable? Does a planar graph if $3$ colorable ...
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1answer
80 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/...
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1answer
110 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 ...
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100 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 ...
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1answer
176 views

Trace of inverse 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}...
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1answer
105 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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1answer
139 views

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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109 views

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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1answer
180 views

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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2answers
315 views

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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1answer
160 views

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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78 views

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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64 views

How to obtain mathematical expectation with the vector as random variable?

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{...
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1answer
95 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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37 views

Lp Wasserstein distance for two distributions which have equivalent partial marginals

Consider two distributions $p(x_1,x_2)$ and $q(x_1,x_2)$ which have equivalent partial marginals, say $p(x_1)=q(x_1)$. I am wondering if there is any relationship between two Wasserstein distances $W_{...
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1answer
180 views

Simple proof of Prékopa's Theorem: log-concavity is preserved by marginalization

The following result is well-known: Suppose that $H(x,y)$ is a log-concave distribution for $(x,y) \in \mathbb R^{m \times n}$ so that by definition we have $$H \left( (1 - \lambda)(x_1,y_1) + \...
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91 views

Concentration inequality for max component of a multivariate Gaussian in the general case

I am looking to bound the variance of the maximum component of a vector distributed multivariate Gaussian in the general case where the Gaussian distribution has arbitrary mean and full covariance ...
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1answer
222 views

Area of $n$-sphere contained outside $\ell_1$ ball

For a given $r>1$, what is the surface area of $\mathbb S^{n-1}$ (the sphere of radius 1 in $\mathbb R^n$) which is contained outside of the $\ell_1$ ball of radius $r$? Or equivalently, if $X\sim ...
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1answer
142 views

Moment generating function of random unit vector

Let $X$ be uniformly distributed on the unit sphere $S^{n-1}$. Is there any result concerning the calculation or bound (particularly lower bound) of $$\mathbb{E}[\exp(X^Tv)]$$ for any $v$?