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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An inequality regarding centered Bernoulli random variables

Let $\left\{V_1,\ldots,V_n\right\}$ be a set of (possibly dependent) identically distributed Bernoulli random variables, with $$ p = \mathbb{P}\left(V_i=1\right) = 1-\mathbb{P}\left(V_i=0\right),\quad ...
Student's user avatar
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3 votes
0 answers
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Using Kac-Rice formula to count average number of sub-regions carved out by $n$ random hyper-planes

Context. This is the first in a set of tiny pieces of a problem I've formulated to help me measure the "complexity" of certain piecewise linear functions. Thanks in advance for your help and patience. ...
dohmatob's user avatar
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8 votes
2 answers
1k views

Talagrand's inequality for the discrete cube

Talagrand showed that if $f$ is a convex $1$-Lipschitz function on $\mathbb{R}^n$, and if $\mu$ is a product of probability measures supported over the interval, then $f$ has Gaussian concentration w....
alesia's user avatar
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9 votes
1 answer
319 views

Visibility in a growing orchard

This is a variant on Polya's orchard problem.1,2 Suppose trees are planted randomly in the plane. The question is: How many trees are visible from the origin as their radii grow? More precisely, ...
Joseph O'Rourke's user avatar
1 vote
1 answer
221 views

Barycenter Map on Wasserstein Space

Let $(X,d)$ be a complete separable metric space, $P_1(X,d)$ be the set of Radon probability measures on $X$ satisfying $$ P_1(X,d)\triangleq \left\{ \nu:\,(\exists x_0\in X)\, \int_{x\in X} d(x,x_0)d\...
ABIM's user avatar
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3 votes
2 answers
615 views

Lifting a probability measure to the power set

Let $X\neq\emptyset$ be a set and let $\mu:{\cal P}(X)\to [0,1]$ be a probability measure. Is there a probability measure $$\bar{\mu}:{\cal P}({\cal P}(X))\to [0,1]$$ with the following property? ...
Dominic van der Zypen's user avatar
7 votes
1 answer
483 views

Understanding Gillman's proof of the Chernoff bound for expander graphs

My question is about the proof of Claim 1 in this paper: Gillman (1993). At the end of the proof, the author says: The matrix product $U^\top\sqrt{D^{-1}}(P+(\mathrm{e}^x-1)B(0)-\mu I)\sqrt{D}U$, ...
Ella Sharakanski's user avatar
4 votes
1 answer
232 views

Operator version of Birkhoff ergodic theorem

Suppose that $(\Omega,\mathcal{E},P)$ is a probability space and suppose that we have a measurable operator $T:\Omega\to\Omega$. Recall that $T$ is said to be egodic if: $T$ is measure preserving: ...
Littlefield's user avatar
11 votes
2 answers
807 views

Computing the sum of an infinite series as a variant of a geometric series

I came across the following series when computing the covariance of a transform of a bivariate Gaussian random vector via Hermite polynomials and Mehler's expansion: $$ S = \sum_{n=1}^{\infty} \frac{\...
Chee's user avatar
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4 votes
1 answer
188 views

Local central limit theorem far from the center

Let $X_i$ be a sequence of iid random variables, $E [X] = 0$, $E [X^2] = 1$ and $E [|X|^k] < \infty$ for some $k \ge 3$. Classical local CLT says that the density function $f_n$ of $\frac1{\sqrt n}...
gregarki khayal's user avatar
4 votes
2 answers
185 views

Uniform convergence of averages for stationary ergodic process

Let $\{X_t, t\in\mathbb R\}$ be a well-behaved$^*$ stationary ergodic process. I'm interested in the uniform convergence of averages: $$ \sup_{|x|\le R_n} \left|\frac1{2n}\int_{x-n}^{x+n} X_t dt - \...
zhoraster's user avatar
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2 votes
1 answer
239 views

Ratio of expectation involving random unit vectors

Let $u=(u_1,...,u_n), v=(v_1,...,v_n)$ be two random vectors independently and uniformly distributed on the unit sphere in $\mathbb{R}^n$. Define two other random variables $X=\sum_{i=1}^nu_i^2v_i^2$, ...
neverevernever's user avatar
6 votes
2 answers
537 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 ...
Upc's user avatar
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1 answer
1k 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{...
Aryeh Kontorovich's user avatar
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0 answers
81 views

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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4 votes
1 answer
1k 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{...
TMM's user avatar
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0 answers
117 views

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 ...
user507474's user avatar
2 votes
3 answers
951 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 $\...
hookah's user avatar
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2 votes
0 answers
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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 $...
πr8's user avatar
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7 votes
1 answer
916 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 ...
JohnA's user avatar
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0 answers
55 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?
Suvadip Batabyal's user avatar
14 votes
1 answer
799 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? ...
Honza's user avatar
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4 votes
0 answers
198 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) ]...
Melika's user avatar
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1 vote
0 answers
104 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 ...
John D's user avatar
  • 11
2 votes
0 answers
108 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. ...
Anindya De's user avatar
2 votes
0 answers
94 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: $$ Z_t = \mathbb{E}\bigg[\int_0^t Y_s ds \bigg| Y_t=k , Y_0=1\bigg] $$ ...
Gabriel's user avatar
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1 vote
1 answer
259 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 ...
Aidan Rocke's user avatar
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3 votes
0 answers
92 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 ...
Vilhelm Agdur's user avatar
2 votes
0 answers
107 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 $\...
dohmatob's user avatar
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3 votes
1 answer
2k 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 ...
user929304's user avatar
1 vote
0 answers
1k 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 $\{0, 1\}$ ...
BenB's user avatar
  • 111
7 votes
2 answers
258 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 ...
Guillaume Aubrun's user avatar
0 votes
3 answers
870 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 ...
Attila Kinali's user avatar
10 votes
0 answers
367 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 ...
喻 良's user avatar
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5 votes
1 answer
166 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,\...
neverevernever's user avatar
3 votes
1 answer
611 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\...
Lwins's user avatar
  • 1,531
6 votes
1 answer
127 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 \...
Elle Najt's user avatar
  • 1,432
5 votes
1 answer
234 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....
Gro-Tsen's user avatar
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2 votes
0 answers
78 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 ...
Thomas Dybdahl Ahle's user avatar
5 votes
0 answers
303 views

Points of continuity of Kullback-Leibler divergence with respect to weak convergence

I know that the Kullback-Leibler $D(\mu||\nu) := - \int_K\log\big(\frac{d \nu}{d \mu}\big) \, d\mu,$ over probability measures on a compact $K$ subset of $\mathbb{R}^d$, is only weakly lower ...
thegain's user avatar
  • 51
6 votes
1 answer
546 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 $L^1(\mathbb{R}^d)$ ...
neverevernever's user avatar
2 votes
0 answers
91 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 > ...
Fran Medjurecan's user avatar
5 votes
1 answer
170 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/...
munchhausen's user avatar
3 votes
1 answer
166 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 ...
poiuy's user avatar
  • 33
4 votes
0 answers
131 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 ...
Daron's user avatar
  • 1,761
7 votes
1 answer
779 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}...
Goulifet's user avatar
  • 2,174
3 votes
1 answer
184 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 \...
dohmatob's user avatar
  • 6,716
6 votes
1 answer
198 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 ...
dohmatob's user avatar
  • 6,716
7 votes
1 answer
358 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 know ...
Daron's user avatar
  • 1,761
13 votes
2 answers
1k 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 ...
Xiaomi's user avatar
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