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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Processes with the same finite dimensional distributions as the solutions to SDEs

Consider a sequence of stochastic processes $\{\tilde{x}^n\}$, $\tilde{x}^n = \tilde{x}^n_t(\omega)$, and Brownian motions $\{\tilde{w}^n\}$. Suppose that for each $\tilde{x}^n$ solves the stochastic ...
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52 views

How to check numerically iterated logarithm law ? (How to choose cutOff lim_n sup_{m: n<= m<= CutOff} ) ?

The law of iterated logarithm asserts that if $x_1,x_2,\dots$ are i.i.d $\cal N(0,1)$ random variables and $S_n=x_1+x_2+\cdots+x_n$, then $$\limsup_{n \to \infty} S_n/\sqrt {n \log \log n} = \sqrt 2, ...
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75 views

Covariance matrix as optimization problem solution?

I have seen the expectation of a random vector expressed as the solution to the optimization problem: \begin{equation} \mathbb{E}[X]=argmin_{v \in \mathbb{R}^n}\mathbb{E}[\|X-v\|_{l^2}^2](:= ...
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74 views

Brownian motion - probability of hitting an open subset of the sphere

Consider a Brownian particle in $\mathbb{R}^n$, starting at the origin. Let $\mathbb{P}_t(A)$ be the probability of the particle striking $A \subset S^{n - 1}$ within time $t$, where $A = \{ (x_1, ...
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1answer
124 views

Transition probabilities for the symmetric random walk on the integers

I found that most references for the symmetric random walk on the integers are for the discrete time case, i.e. the ones that gives us explicit transition probabilities. Now, I am looking at a random ...
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1answer
212 views

Supremum of a martingale

Let $(X_n)$ be a martingale. What can be said about the distribution of its maximum over a window of fixed length: $$M_n = \max_{n-10 \leq k \leq n} X_k$$ or about the "range" over a window: $$R_n = ...
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131 views

Linking Wasserstein and total variation distances

I seek to bound the total-variation distance between two probability measures $p_1$ and $p_2$. It is extremely easy to build a parameter space where $p_1$ and $p_2$ are the marginals of some joint ...
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177 views

What is the formal name of this set-related concept?

I "invented" a concept and it feels like it has already been invented before. I would like to know whether such a concept exists and if so, what is its name? Let $S$ be a family of finite sets. ...
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176 views

Gaussian and the convex hull of moment curves

Let $c_1,\dots, c_d$ be the first $d$ moments of the standard normal distribution. Does the point $(c_1,\dots, c_d)$ lie in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$, for a ...
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95 views

weak-* versus entropy growth

General question. Let $\eta_{n}$ be a sequence of invariant measures on $\{0,1,2,...,p-1\}^{\mathbb{N}}$ and $B$ the Bernoulli uniform measure. Knowing that $\eta_{n} \rightarrow B$ in the weak-* ...
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132 views

Moment matching: construction of a mixture of Gaussian distribution with lower moments identical to Gaussian

This is a question related to the statistical model behind independent component analysis (ICA). We assume that $Z \sim N(0,1)$. Our goal is to construct a random variable $X$ that follows a ...
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56 views

Circular process ergodic?

Let us define a continuous-time Markov process on a circle consisting of $m-$ equally spaced points, i.e. every point has two neighbours. Now, we define a space of functions $S:= ...
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1answer
85 views

Weak convergence of process

Background: I am trying to compute the weak limit of the following model from mathematical biology that is supposed to exist: Let $$L(f)(\eta)= \sum_{x \in \mathbb{Z}}\frac{1}{2}\left(1_{\eta(x+1) ...
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52 views

Simulate a graph from a certain distribution

I am wondering if anyone can indicate whether the following is a solved problem. I don't care about time of the algorithm currently. Consider a general probability distribution F on simple graphs ...
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83 views

Malliavin differentiability of solutions to SDEs

In Bass's book on Diffusions and Elliptic Operators, the author gives a brief introduction into Malliavin Calculus. He calls a functional $F:C([0,1],\mathbb{R})\rightarrow \mathbb{R}$ $L^p-$smooth if ...
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103 views

Almost independent Bernoulli variables

There is some global parameter $n\to\infty$. And a function $N=N(n)\to\infty$. Let $X^n_1,X^n_2,\ldots,X^n_N$ be independent Bernoulli random variables, where $\delta\le P(X^n_i=1)=1-P(X^n_i=0)\le ...
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1answer
249 views

Is this a log-concave function?

Let $(a_k)$ be a log-concave positive decreasing sequence. Is $\sum\limits_{k=1}^n a_k(1-e^x)^{k-1}$ log-concave in $x<0$, for each natural $n$?
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129 views

Weak convergence in random measures

I don't understand the following as I read along a proof in a paper (Page 66, "Asymptotic Behaviour of some interacting systems", by Sylvie Meleard): We denote by $\mathcal{P}({M})$ the space of ...
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1answer
94 views

Do we have independence if we let the indices of the events increase?

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Consider events indexed by $m, n \in \mathbb N$: $ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent. ...
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216 views

Asymptotics of functional of i.i.d. Rademacher random variables

Let $X_1,\ldots, X_n$ be i.i.d. Rademacher random variables. That is, $\operatorname{Pr}(X_i = 1) = \operatorname{Pr}(X_i = -1) = 1/2$. I was wondering if the following argument is true: $$ \mathbb{E} ...
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206 views

Batched Coupon Collector Problem

The batched coupon collector problem is a generalization of the coupon collector problem. In this problem, there is a total of $n$ different coupons. The coupon collector gets a random batch of $b$ ...
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48 views

Regularity of the entrance measure of SRW

Let $S(n)$ be the discrete sphere of radius $n$ (i.e., the internal boundary of the Euclidean discrete ball $B(n)$) centered in the origin, and consider a simple random walk starting at some ...
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91 views

Is a local martingale with constant expectation necessarily a martingale?

Suppose $X\in \mathbb R$ is a weak solution to the SDE $dX_t = \sigma(X_t)dW_t$, in which $W$ is a one-dimensional Brownian motion, and $\sigma$ is Borel measurable so that a weak solution exists and ...
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244 views

Are $\left[\begin{matrix}x_\ell \\ x_\ell\varphi_k^\ell\end{matrix}\right]$ linearly independent?

Let $\varphi_k\in\mathbb{C}$ be a primitive $k$-th root of unity, and define the sets ...
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190 views

concentration inequality for entropy from sample

Consider a measure $\mu$ on a finite set, and let $x_1, \ldots, x_n$ be i.i.d samples from $\mu$. Then the expression $S_n = -\frac{1}{n} \sum_{i=1}^n \log \mu(x_i)$ converges by a.s. to the entropy ...
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1answer
122 views

Approximation of the cumulative normal distribution

As is well known, there is no explicit formula for $\int_{-\infty}^\infty step(t−x)\cdot e^{−t^2/2}dt=\int_x^\infty e^{−t^2/2} dt$ for generic $x,$ where $step(z)$ is the step function, $step(z)=1$ ...
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1answer
89 views

Shift Invariance of Backward Martingales for tail trivial probability measures

Consider the infinite cartesian product $\Omega=\{0,1\}^{\mathbb{N}}$ as a measurable space endowed with the $\sigma$-algebra $\mathscr{F}$ generated by the cylinder sets and $\sigma:\Omega\to\Omega$ ...
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2answers
206 views

Why are the vectors with this special structure linearly independent with high probability?

Given $a_i\in\mathbb{R}^m$ for $i=1,\ldots,2m$ with independent and identically random entries of some continuous distribution. Every choice of $m$ vectors from $\{a_1\ldots,a_{2m}\}$ is linearly ...
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1answer
92 views

Distribution of maximum unique number of several random numbers

Suppose discrete random variables $\{X_1, X_2, ..., X_n\}$ are i.i.d. described by the probability function: $f(x) \equiv \text{Pr}(X_i = x)$, and $X_i \in \{1,2,3, ..., m\}$. Let $Y$ be the ...
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113 views

Expectation of a specific random variable on the probability space of $n\times n$ matrices over $\{0,1\}$

Let $\mathcal{G}_{n,\frac{1}{2}}$ be the probability space of $n\times n$ matrices over $\{0,1\}$ and each entry of the matrix is independently equal to 1 with probability $\frac{1}{2}$ and equal to 0 ...
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1answer
163 views

A lottery on coins in a convex set

You play the following game. You get $4n$ gold coins and have to arrange them in the unit square in general position (no two coins have the same x or the same y coordinate). Call this set of coins ...
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171 views

Brownian motion in $\mathbb{R}^n$, probability of hitting a set

Consider a particle undergoing Brownian motion in $\mathbb{R}^n$, starting at the origin, and let $B(t)$ denote its position at time $t$. Let $X$ be an arbitrary subset of $\mathbb{R}^n$. I am trying ...
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42 views

Exploiting conditional independence for inference in Bayesian networks

How is conditional independence used for making probabilistic inference in Bayes networks easier or more efficient? For example, given the following Bayes network: Let's say I want to compute ...
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1answer
57 views

Local extrema of a posterior probability

Let $x$ be a binary random variable and $z$ be an arbitrary random variable. $x$ and $z$ are, in general, not independent. Let $y_1, \ldots y_n$ be $n$ identically distributed binary random variables ...
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62 views

Order statistic of Markov chain sample path and related probabilities

Consider a 1D sample path, denoted as $\{X(1), ..., X(t), ..., X(n)\}$, generated from a discrete time finite state (time homogeneous) Markov chain over states $\{1,...,m\}$, with transition ...
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164 views

Does walk on $Z^d$ with steps $(\pm 1,\pm1,\ldots,\pm 1)$ return to origin?

If the steps are iid uniform as in the title, is the return probability known? Is it positive? Answers, comments, references welcome. Clearly each of these steps is not equivalent to $d$ steps of type ...
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149 views

Multivariate CLT with varying dimension size

If $X_i$ is a sequence of $d$ dimensional i.i.d. integer valued random vectors with covariance matrix $\Sigma$ and $\mathbb{E}(X_i) = \mu$. Let each element of $X_i$ be chosen i.u.d. from $\{-1,1\}$. ...
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84 views

Question about the weak convergence of probability

Let $\mu$ be a probability measure on $\mathbb R$ and set $$c(K):=\int_{\mathbb R}(x-K)^+d\mu(x).$$ Assume that one has a sequence of probability measures $(\mu_n)_{n\ge 1}$ s.t. $$\int_{\mathbb ...
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54 views

Definiteness and infinite divisibility of kernels including heat semigroup

Let $P_{t}$ be the usual heat semigroup. Can one show (preferably) or disprove that for arbitrary $k \in \mathbb{R}_{>0}$ and $n \in \mathbb{N}$ we have \begin{equation} ...
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1answer
115 views

Question abouth Skorokhod representation of random variables (II)

This is a continuation of Question abouth Skorokhod representation of random variables Let $\mu$ and $\nu$ be two probability measures on $\mathbb R$ such that $$\int_{\mathbb R}|x|^pd\mu(x),~ ...
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1answer
108 views

Hellinger integral for the Student/Cauchy family

Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral is $H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$. Let now $p$ be ...
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110 views

Question abouth Prokhorov metric

Let $X$ and $Y$ be two random variables with first order moments, i.e. $E[|X|]$, $E[|Y|]<+\infty$. Assume further that $$E\left[|X-Y|\right]<\varepsilon.$$ Set $Law(X)=\mu$ and $Law(Y)=\nu$, ...
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108 views

Maximal Correlation versus Correlation Coefficient When one RV is Gaussian

Let a pair of random variables $(X,Y)$ be continuous random variables (i.e., they both have density with respect to Lebesgue measure) with joint distribution $P_{XY}$. The maximal correlation ...
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67 views

Conditions which might imply a function is independent over its arguments

Suppose $f: [0,1]^{k+1} \to [0,1]$ and let $(a_0, \dots, a_k, b)$ be a tuple of $[0,1]$-valued random variables. Suppose for each $0 \leq i \leq k$ there is a collection of tuples $$ A_i = ...
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148 views

Groups with probability measures

Are there algebraic structures that integrate groups with probability measures? For instance, can the closure operation on a group be assigned a probability that says "how much" a member belongs to ...
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1answer
87 views

Negative population variable importance

I asked this question on stats.stackexchange and even elsewhere, but it never received an answer. I just state the probabilistic problem here. It is about the optimality of the conditional ...
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124 views

divisibility of uniform distribution [closed]

Let $X$ and $Y$ be independent and identically distributed random variables. Can $X+Y$ be a uniform distribution? (Please prove.) In other words, is a uniform distribution divisible? The meaning of ...
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69 views

Conditional version of martingale difference concentration inequality

Let $M_n$ be a $\mathscr{F}_n=\sigma(\eta_m,\theta_m, m\leq n)$ measurable martingale difference sequence. Then is it possible to find a exponential tail bound for the following $$P(|M_{n+1}| > ...
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111 views

Interplay between CLT and convergence in Total Variation

Given a random variable $X$ with bounded moments such that $E[X] = 0, E[X^2] = 1$, let $F_n$ denote the distribution $\sum\limits_{i=1}^d\frac{X_i}{\sqrt{n}}$ where each $X_i$ is an independent copy ...
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Comparison between the entrance measure and the harmonic measure

Consider the standard two-dimensional Brownian motion, and define $\tau(A)$ to be the hitting time of $A\subset \mathbb{R}^2$. Let $hm_A$ be the harmonic measure (from infinity) on $A$. Let $B(r)$ be ...