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

learn more… | top users | synonyms (1)

-1
votes
1answer
83 views

Proving maximal entropy [closed]

It is quite easy to prove that $$H(S) \leq \log_2(|A|),$$ where $A$ is the number of events, using the Jensen inequality $$H(S) = E_S[\log_2(\frac{1}{P_S(s)})]\leq \log_2(E_S[(\frac{1}{P_S(s)})]) ...
3
votes
2answers
108 views

Expected value of Bernoulli quadratic forms

Let $\mathbf{Y}\in\mathbb{R}^{n\times n}$ be a symmetric matrix. Let $\mathbf{x}\in\mathbb{R}^n$ be random vectors with entries i.i.d. $\pm 1$ with equal probability. I'm interested in a lower bound ...
3
votes
1answer
106 views

Sharpened Pinsker inequality for special case

Let $B(p)$ denote the Bernoulli distribution over $\{0,1\}$ and $B(p)^n$ the corresponding product distribution over $\{0,1\}^n$. For $n>1$ and $0<x<1$, define $$P_n(x):=B(\frac12+\frac ...
0
votes
1answer
115 views

Counterexample: weak convergence doesn't imply $L^1-$convergence [closed]

I'm not sure my question is of research level, but I cannot find the answer in the existing reference. Let $\mu_n$ be a sequence of probability measures on $\mathbb R$ satisfying $$\int_{\mathbb ...
1
vote
1answer
114 views

Integration against Borel measures on compact Hausdorff spaces

I am studying the properties of integration against Borel measures and Baire measures. And I am not sure whether the following proposition is correct and I tried to give a proof. Suppose that $X$ ...
2
votes
0answers
113 views

Throwing darts at a barn and putting a bullseye around them in higher dimensions

Let $X \in \mathbb R^d$ be a large domain (a ball of radius $r$ for $r$ large should suffice) Let $B$ be a ball of radius $1$. Consider the ratio $$ \frac{ \left| \left\{ x_1,\dots,x_n \in X \mid ...
14
votes
0answers
145 views

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 ...
4
votes
0answers
54 views

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: $M f(X) := \sup_{\rho \in [0,1]} \mathsf{E} [f(\rho X + (1-\rho^2)^{1/2} X^\prime) \mid X]$ ...
2
votes
1answer
119 views

Question on Wiener processes not hitting 0

Let $W_t$ be a standard Wiener process, and $0\leq a < b$. Let $\hat{W}_t:=W_{a+t}-W_a$. Then $\hat{W}_t$ is also a standard Wiener process. I think that the following should be true: $$\mathbb ...
6
votes
1answer
314 views

Liouville property - a very basic question

Let $\mathbb{F}_2$ be the free group on two generators. By a result of Kaimanovich and Vershik, for each measure $\mu$ on $\mathbb{F}_2$ such that the support of $\mu$ generates $\mathbb{F}_2$, we ...
1
vote
0answers
34 views

Alternative to generic chaining bounds for a particular family of stochastic processes

Generic chaining provides a general but rather abstract framework to bound suprema of stochastic processes. In many applications, however, we know more about the expression of the stochastic process. ...
5
votes
0answers
71 views

Probabilistic distribution of sandpile model type

Let $G=(V,E)$ be a connected graph. Assume that $m\leqslant |V|$ hedgehogs sit in the vertices of $G$. If there are $r\geqslant 2$ hedgehogs in the same vertex $v\in V$, one of them goes to a randomly ...
0
votes
1answer
86 views

Equal probability of having even/odd number of ones in many Bernoulli trials with different probabilities? [closed]

This problem has probably been solved somewhere but I could not find it. We have $n$ Bernoulli random trials $X_i$ with different occurrence probabilities, $\mathrm{Pr}[X_i=1]=p_i>p_{\min}>0$ ...
1
vote
0answers
41 views

How to estimate the size of balanced biclique in random bipartite graph?

We have a random bipartite graph $G=(V,U,E)$ and $|V|=|U|=n$, in which any vertex pair $<v,u>$ ($v\in V$,$u\in U$) exists an edge with probability $p$. A balanced bipartite complete graph is a ...
12
votes
2answers
256 views

Shortest path through $n^{1/3}$ points out of $n$

Say I sample $n$ points uniformly at random in the unit cube in $\mathbb{R}^3$, and then I look for the shortest path through $n^{1/3}$ of those points (rounding up, say). What happens to the length ...
0
votes
1answer
100 views

How to calculate the expected value of complex-valued random variable? [closed]

Suppose $\theta_1,\theta_2,\cdots, \theta_n$ are independent and identically distributed (i.i.d.) real-valued random variables and here we specifically consider the uniform distribution in the ...
2
votes
1answer
113 views

About Renyi entropy

If one is given a joint probability distribution over a finite set of discrete random variables then I guess there a notion of $\alpha-$Renyi entropy defined for it as $S_\alpha (X_1,..,X_n) = ...
2
votes
1answer
110 views

Averages of vector inner products over the Haar measure

Consider arbitrary unit vectors $w,x,y,z \in \mathbb{C}^d$. Is there an explicit formula for what this average is? $$ \int \mathrm{Tr}( \psi \psi^* \, \, w x^* \,\, \psi \psi^* \,\, y z^*) d\psi $$ ...
1
vote
1answer
89 views

Limit (Convergence) of stopping times

Let $B=(B_t)_{0\le t\le T}$ be a continuous semi-martingale and $\mathbb F=(\mathcal F_t)_{0\le t\le T}$ be its natural filtration. Denote by $\mathcal C_b(\Omega\times \mathbb R_+)$ the space of ...
7
votes
1answer
196 views

Tightness and Functional Analysis

Let $(\Omega , \mathbb{P})$ be a probability space and $X$ be a real-valued random variable. Then we immediately have the push-forward measure $\mu$ on $\mathbb{R}$ and one can think of $\mu$ as an ...
1
vote
0answers
69 views

Finding a closed form for a certain double integral

I am working with bivariate and accurate Birnbaum-Saunders distribution to find the probability density function of a particular model for this, I would like to find a closed form for the full below: ...
1
vote
0answers
34 views

What is meant by local time of BM on the boundary $\partial D$?

I'm familiar with local time $L_t^a$ at level $a$ for a 1-D Brownian motion $B$. I'm reading this paper which talks about a 2D Brownian motion $B$ in a bounded domain $D$ that gets reflected when it ...
0
votes
0answers
41 views

Brunett Derrida behaviour for the branching brownian motion with selection

In this paper Berard and Gouéré proved that for a binary branching random walk with selection of the N rightmost particles the cloud of particles moves asymptotically at a deterministic velocity ...
1
vote
1answer
148 views

Averaged geometric series with floor function

Given a value $p\in[0,1]$ (a probability of occurrence), I would like to bound the following expression: $$ s\frac{1-(1-p)^{k+1}}{p(k+1)} + (1-s)\frac{1-(1-p)^{k}}{pk},\ \ \ \text{where $k=\lfloor ...
2
votes
0answers
64 views

Question about martin boundaries of random walks induced on transient subgroups

Suppose $\Gamma$ is a discrete, finitely generated, non-amenable group, and consider a random walk given by a measure $\mu$. Assume the measure is symmetric, finitely generated, and the support of ...
0
votes
0answers
175 views

A contractive mapping which I don't understand

Given a matrix $Y$ and a vector $c$ define the following iteration $\hat{c} = f(c)$, where each element of $\hat{c}$ is given by $$\hat{c}_{\ell} = \frac{\sum_k ...
5
votes
1answer
310 views

Does every (generalized?) Markov chain admit transition probabilities?

To pose the question let us start by recalling the following notions: Transition Probabilities. A transition probability matrix between two measurable spaces $(S,\mathcal{S})$ and ...
-1
votes
1answer
115 views

Is Gaussian the unique 2-stable distribution? [closed]

It is well known that Gaussian distribution is a 2-stable distribution. (For more information about p-stable distribution, please refer to Stable Distribution.) But is Gaussian the unique 2-stable ...
1
vote
0answers
49 views

variance of log of ratio of chi-square variables

Let X be a chi-square variable with two degrees of freedom. Let A and B be to arbitrary constants, with $A>B>0$. I need the variance of $Y=\log(1+AX)-\log(1+BX).$ The mean is, maybe not ...
3
votes
0answers
58 views

Algorithm to calculate moments of uniform distribution on convex polyhedra

There is system of linear inequalities $$ Ax \leq K, $$ $$ x\geq a, x\leq b. $$ $A$ is $(n\times m)$-matrix, where $n\approx 100$ and $m\approx 10000$, $rank(A)=n$. Suppose that on set of solutions ...
0
votes
0answers
48 views

stochastically decreasing sequence converges in distribution

Let $(X_i)_{i=1}^\infty$ be independent nonnegative integer valued random variables. Suppose that $X_n \succeq X_{n+1}$ (in the stochastic dominance sense). Does it follow that $X_n \overset{d}\to X$ ...
1
vote
2answers
186 views

High order central moments of a symmetric binomial variable

Consider a random variable $X\sim B(n,\frac 12)$. I'm trying to estimate the asymptotic behaviour of its central moments $E((X-\frac n2)^r)$, where $r$ is even and in the range $\Omega(1)\leq r\leq ...
2
votes
2answers
150 views

Do we have Karhunen–Loève expansion for White Noise?

Let $W$ be a random process (my White Noise) on $[-1,1]$ such that: $W(t)$ is a normal random variable with mean $0$ and standard deviation $1$ for all $t \in [-1,1]$ $E(W(t)W(s)) = 0$ for all $t, s ...
1
vote
0answers
56 views

Norm-averaging reference request

(Apology in advance for the broadness of this question) I recently came across a relatively simple application where I needed to "balance" the "spreaded-out-ness" of a function with the "peaked-ness" ...
7
votes
0answers
184 views

Can primes be (almost) random sequence in von Mises sense?

Random models for primes (such as Cramer's model) have been extensively used for informal justification of various conjectures involving primes. It is crucial to understand in what sense sequence of ...
1
vote
1answer
37 views

Is there an easy way to convert a non-deterministic optimal policy to a deterministic optimal policy for a given MDP?

For a MDP (Markov Decision Process) is there an easy way to convert a non-deterministic optimal policy into a deterministic optimal policy? The trivial way will take ...
0
votes
0answers
256 views

Probabilistic inequality problem: how do I find its least upper bound function?

Before posing the question itself, it is indispensable to give the definition from which it arises. First of all, let's restrict our attention to the vectors $\overrightarrow{x} = ...
1
vote
1answer
40 views

Subquadratic multiplication of probability mass functions (with log-convolution?)

We are currently looking for a fast, i.e. subquadratic, algorithm for the following equation: $z_m = \sum_{i,j :\, (i \cdot j) = m} x_i \cdot y_j$. That is, we are given two finite input vectors $x$ ...
1
vote
1answer
150 views

connection between the statistical properties of a scalar field and its columns

Consider a scalar field $s:[0,1]^3 \to \mathbb{R}$ and its "column" field \begin{equation} c: [0,1]^2 \to \mathbb{R}: (x,y) \mapsto \int_0^1 s(x,y,z) \,\mathrm{d}z. \end{equation}. What can be said ...
6
votes
1answer
118 views

Lowest index giving half of the sum

Numbers $x_1,x_2,\ldots,x_n$ are drawn independently and uniformly from the interval $[0,1]$. Order them as $y_1\ge y_2\ge\dots\ge y_n$, and let $S$ be their sum. Let $k$ be the smallest index such ...
9
votes
1answer
446 views

combinatorics on cyclic sequences

Given $m\geq 1$, let $I=(a_1,\ldots,a_{3m})$ be a sequence such that $I$ contains exactly $m$ zeros, $m$ ones, and $m$ twos. Given $i=1,2$ and $j\leq 3m,k\leq m$ we can define ...
0
votes
0answers
28 views

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: ...
2
votes
1answer
149 views

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 ...
1
vote
1answer
327 views

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 ...
0
votes
0answers
51 views

limit multiple integral

I want to know if $\lim_{T-> \infty}$ of this integral $$ \frac{\sigma^{4}C_{H,K}^{2}}{4 T^{4HK}e^{2\theta T }}\\ \times ...
3
votes
1answer
281 views

Bounds on $\int \log(1+x) g(x) \mathrm{d}x$?

Let $X$ and $Y$ be two continuous real random variables with common support $(0,x_{\max}]$ and with PDF $f_X(x)$ and $f_Y(y)$. Assume that $\Pr [Y\geq\beta \mid X<\beta] \leq k$ and that $\Pr ...
2
votes
0answers
151 views

Must rows of a transition matrix be distinct?

Is it true that for all continuous time Markov processes on a countable state space $S$, we have all rows of the transition matrix $\mathbf{P}_t$ are distinct for all time $t\in[0,\infty)$ ? This ...
3
votes
0answers
58 views

Most visited vertex in a random walk with place dependent drift

Consider the following Markov chain on $\mathbb{Z}$: $$ P(x,x+1)=1-P(x,x-1)=\frac{1}{2}+e^{-|x|}\cdot \mathbf{1}_{\{x\neq 0\}} $$ Do there exist constants $c,C>0$ such that $$ c\cdot P^t(z,z) ...
1
vote
0answers
34 views

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 ...
2
votes
0answers
108 views

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 ...