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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
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1answer
18 views

Union of random intervals with total length equal to infinity

Let $a_1,a_2,\dots$ be a sequence of positive numbers less than 1, such that $$\sum_{n=1}^\infty a_i= \infty,$$ and $S^1 = \mathbb{R}/\mathbb{Z}$. Suppose $I_1,I_2,\dots$ be random intervals with ...
-3
votes
1answer
40 views

How to compute the conditional probability of P(X|X^2) [on hold]

If X ~ N(0, 1), How to compute the conditional probability of P(X|X^2), is it equal to 1?
3
votes
0answers
25 views

Is the maximum of independent Poisson random variables log-concave?

Let $X_1,\ldots, X_n$ be independent Poisson random variables with parameter $\lambda > 0$. Define $$M_n=\max \lbrace X_1.\ldots, X_n \rbrace,\,\,p_j=\mathbb{P}(M_n=j).$$ Is it the case that $M_n$ ...
3
votes
1answer
105 views

about an interesting moment generating function

Let $X_1,\ldots,X_n$ be iid Rademacher variables, i.e., $P(X_1=1)=P(X_1=-1)=1/2$. CLT says that $Y_n\equiv \sqrt{n}\bar{X}$ converges in distribution to $N(0,1)$ as $n\to\infty$. So $Y_n^2$ is ...
3
votes
1answer
89 views

What is the probability for a Binomial to be greater than other?

Let $X = B(n, p)$ and $Y = B(k, q)$ be two random variables with binomial distribution and, let $s$ be a positive integer. Assume that $n > k+s$ and $np \geq kq+s$. What is the probability for $X$ ...
4
votes
1answer
144 views

Is this a random walk? Does it have a name?

By combining two methods I've stumbled into a rather messy random walk situation. I have the typical random walk setup $$\theta_{i+1} = \theta_{i} + \hat{\theta}_{i+1}$$ Where $\hat{\theta}_{i+1} \...
2
votes
0answers
34 views

Can we transform $\int_\rho^1 (W_t - W_{t-\rho}) dW_t$ to make its law $\rho$-invariant?

I just bumped into the stochastic integral $$ \int_\rho^1 (W_t - W_{t-\rho}) dW_t $$ where $0 < \rho < 1$ is a constant and $W$ is a standard Wiener process. It would be nice if we have a closed-...
2
votes
0answers
29 views

Continuous Local Martingales under time change under what conditions are they still local martingales?

This question is motivated by reading a section in Continuous Martingales and Brownian Motion by Daniel Revuz, Marc Yor. In Chapter V there is a section on time-change: Definition: A time change $C$...
3
votes
0answers
85 views

gaussian upper bound on spherical heat kernel

It is known that the heat kernel on n-sphere satisfies $p_t(x,y)\leq Ct^{-n/2}e^{-d(x,y)^2/5t}$ for all $t\in (0,T)$. Can something be said about how big C needs to be?
4
votes
1answer
72 views

KL divergence and mixture of Gaussians

Do we have an exact formula to compute the KL divergence between 2 mixtures of Gaussians (i.e convex combinations of a finite number of Gaussian distributions)? If not exactly known, are there good ...
6
votes
1answer
139 views

Coverage of balls on random points in Euclidean space

We have n points randomly distributed in a d-dimensional unit hypercube. We randomly sample k of those points and center a ball with radius r on each of those k points. Does there exist an estimate of ...
-1
votes
0answers
29 views

Relation between significance level and sample size

Suppose $X_1, \ldots, X_n$ is a random sample of size $n$ from some Bernoulli distribution. Consider the hypothesis testing problem $H_0: p= p_0$ vs $H_1: p=p_1$ with significance level $\alpha$, that ...
5
votes
1answer
158 views

Approximation of Wasserstein distance between $p_\theta$ and $p_{\theta + d\theta}$

Given a parametric family of distributions $\{p_\theta|\theta \in \Theta\}$, one can show that under some regularity conditions, the following approximation is valid $$\operatorname{KL}(p_\theta\| p_{...
1
vote
1answer
123 views

A simple two variable analytic inequality, inspired by probability

I'm trying to prove the following inequality: $$ bf_1g_1 + (x-b)f_1g_0 + (y-b)f_0g_1 + (1-x-y+b)g_0f_0 \le (|f_1|^p x + |f_0|^p (1-x))^{1/p} (|g_1|^p y + |g_0|^p (1-y))^{1/p} $$ where $0\le xy\le b\le ...
0
votes
1answer
50 views

Is the law of $\sup_{l \leq t \leq u} \frac{|B_t|}{\sqrt{t}}$ atomless?

Let $B_t$ denote a standard Brownian motion, and $0 < l < u$. I am wondering if the law of $\sup_{l \leq t \leq u} \frac{|B_t|}{\sqrt{t}}$ is atomless, that is, $\mathbb{P}\left(\sup_{l \leq t \...
2
votes
1answer
185 views

Schwartz space on $\bigcup_{n=1}^CR^n$

I have an application where I need to work with the following idea. Let the space $\bigcup_{n=1}^C \mathbb{R}^n$ be associated with the metric $d$ such that for $x=(x_1,\cdots,x_n)$ and $y=(y_1,\cdots,...
2
votes
2answers
53 views

Draw samples from distribitions in the neighborhood of a fixed distribution

Disclaimer Sorry in advance for vagueness. I'm still trying to get my ideas right on this one. Setup So, let $P$ be a distribution on a Euclidean space $X$ with an $\ell_p$ metric, and let $P_\...
3
votes
0answers
30 views

Cadlag and adapted (usual conditions assumed) imply progressively measurable (related to Protter's Stochastic Calculus theorem 6)

Hi maybe someone on here can help me. I have been stuck on showing this fact for several months. I asked this question in the stack exchange and it has floated around for a while but to no avail. I ...
2
votes
1answer
61 views

Improving equi-integrability for a family $\mathcal F$ in $L^1(\Omega)$

Let $\mathcal F$ be a weakly compact subset of $L^1(\Omega)$. Dunford–Pettis theorem says that $\mathcal F$ is uniformly integrable. Also, by de la Vallée-Poussin theorem we can find an increasing ...
2
votes
0answers
50 views

A conjecture characterizing almost uniform convergence of finitely additive conditional probabilities

This question is a continuation of a question I asked a couple weeks ago. Let $(\Omega, \mathcal{C})$ be the Cantor space of binary sequences equipped with the usual product topology, and let $(\...
1
vote
1answer
89 views

Giving Uniform Bound on Differences of Sums of Converging Polynomials

The title does not quite capture the essence of the difficulty, please allow me to be more explicit here. I thought of this question when I was trying out an open problem by Ovidiu Furdui(See problem ...
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vote
0answers
31 views

Random solute transport equation

After reading the article Probability density functions for solute transport in random fields, by M. Shvidler, K. Karasaki, see https://pubarchive.lbl.gov/islandora/object/ir%3A116072/datastream/PDF/...
5
votes
2answers
364 views

Existence of Solution, System of Equations

Suppose $P(\lambda, i)$ is the probability that a Poisson random variable with average $\lambda$ is equal to $i$, i.e. $\frac{\lambda^i}{e^{\lambda}i!}$ I think the following system of equations ...
4
votes
1answer
177 views

$Pr(A>B)$, where $A$ and $B$ are sum of Bernoullies

Let $X= x_1 + x_2 + \ldots + x_m$, $Y=y_1 + y_2 + y_3 + \ldots + y_n$, and $Y' = y'_1 + y'_2 + \ldots + y'_n$, where Each $x_i$ is a Bernoulli variable which takes value $1$ with probability $p_i>...
2
votes
1answer
93 views

How to estimate a total variation distance?

Let $X_1, \ldots, X_n$ be independent Bernoulli random variables. Then $Pr[X_i=1]=Pr[X_i=0]=1/2$. Let $X = (X_1, \ldots, X_n)$ and $v \in \{0,1\}^n$, $Y=v \cdot X$, $Z=Y-1$. Let \begin{align} \mu_1(x)...
0
votes
0answers
36 views

Dominating powers of a random matrix

Let $A_n$ be a (sequence of) random matrix such that $ A_n = (a_{ij})_{1 \leq i,j \leq n}$ and the $a_{ij}$ are iid, $\mathbb E\left[ a_{ij}\right] = 0 \quad E\left[ \vert a_{ij}\vert^2\right] = 1$. ...
1
vote
0answers
55 views

If $f$ is a measurable random field, then $(ω,x)↦E[f(x)\mid F](ω)$ has a measurable version $g$ and $E[f(X)\mid F]=g(X)$ for all $F$-measurable $X$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathcal F\subseteq\mathcal A$ be a $\sigma$-algebra on $(\Omega,\mathcal A)$ $(E,\mathcal E)$ be a measurable space $f:\Omega\times ...
1
vote
0answers
27 views

Differentiability of a stochastic process depending on a spatial parameter

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $T>0$ $I:=(0,T]$ $d\in\mathbb N$ $M:\Omega\times\overline I\times\mathbb R^d\to\mathbb R$ such that $M(\;\cdot\;,\;\cdot\;,x)$ is $\...
2
votes
0answers
66 views

Width of symmetric groups

MSE crosspost For any (finite) group $G$ its length $l(G)$ is the length of maximal chain of proper subgroups (it's known and pretty widely used invariant). But we can also define width function $w_G(...
-1
votes
0answers
32 views

Distribution of fractional parts generated via iid random variables

I am interested in understanding the following setup (if it makes any sense at all). Let $X_1,X_2,\dots,X_n,\dots \sim P$; i.i.d. integer-valued random variables, where $P$ is a probability ...
9
votes
1answer
243 views

Is KL divergence $D(P||Q)$ strongly convex over $P$ in infinite dimension

By KL divergence I mean $D(P||Q) = \int dP \log(\frac{dP}{dQ})$. I am looking for the conditions under which this strong convexity is true and possible references. I could not find an answer for ...
3
votes
1answer
61 views

Lindeberg implies convergence of max of conditional variances in L1

The following is taken from Dvoretzky, 1972, ASYMPTOTIC NORMALITY FOR SUMS OF DEPENDENT RANDOM VARIABLES, Equation 4.6. $$\{X_{n,k}\}_{n=0,1,...;k=0,1...,k_n}$$ is a (triangular) array of r.v.'s /w ...
2
votes
1answer
131 views

*Full proof* references for Markov generators with various boundary conditions

(Note: I've migrated this question from math.stackexchange, as the lack of answers there made me believe it was perhaps too advanced for that forum.) Consider the one-dimensional heat equation $$\...
0
votes
0answers
21 views

Why do middle roots of the $\chi(p)$ graphs and percolation thresholds vary linearly with diagonal probability $q$ (in large random binary matrices)?

Consider a large $N\times N$ square lattice, where each cell has a probability $p$ of being "occupied" (let's call denote them as "black") and a probability $1-p$ of being empty (let's denote them as "...
2
votes
1answer
66 views

Generalization: (The “number” of) smaller sized clusters in large random binary matrices follow a descending order. Why?

This is a sequel to the question: Why is number of single cell clusters always greatest in a random matrix? In their answer, @Aaron Meyerowitz came up with a nice strategy to prove why the number of ...
3
votes
1answer
117 views

Why is number of single cell clusters always greatest in a random matrix?

Consider a large $N\times N$ square lattice, where each cell has a probability $p$ of being "occupied" (let's call denote them as "black") and a probability $1-p$ of being empty (let's denote them as "...
3
votes
1answer
206 views

Approximating the expectation of a matrix inverse

Let $$R := A \Lambda^{-1} A^H + \frac{1}{\gamma} I_n$$ where $A$ is a given $n \times m$ matrix (where $m \gg n$), $$\Lambda := \mbox{diag} \big( \lambda_1, \lambda_2, \dots, \lambda_m \big)$$ ...
0
votes
0answers
48 views

Question about Protter's proof of the Ito's formula

The following is a question about a notation that Protter uses in the proof of the Ito's formula for cadlag processes of finite variation (FV) that appears on Stochastic Integration and Differential ...
3
votes
1answer
83 views

Reference Request: Simple Random Walk on $\mathbb Z$ is Unimodal

I am looking for a reference to the following claim. Let $X = (X_t)_{t\ge0}$ be a continuous time simple random walk. Then $$ m \mapsto P(|X_t| = m) : \mathbb N \to [0,1] $$ is (weakly) decreasing (or ...
1
vote
0answers
52 views

Hoeffding's inequality for random vectors

Let $x_1, \ldots, x_n$ be $n$ i.i.d. samples of a bounded random variable $X \in [a, b]$. We know from the Hoeffding's inequality that : $$\mathbb{P} \left( \left| \frac{1}{n} \sum_{i=1}^n x_n - \...
1
vote
1answer
81 views

are there measure preserving mapping in this case?

Suppose f and g are two Borel function on [0, 1]. The push-forward of the Lebesgue measure on [0,1] by f and by g are the same. Then are there some Borel measurable function from [0,1] to [0,1], ...
0
votes
0answers
37 views

Integral of the product of Normal density (PDF) and CDF with limits

Similar to a previous post, I need to integrate the product of a density (PDF) and a CDF, but this time in just the nonnegative domain. My equation is of the form: $$\int_{0}^{\text{∞}}\Phi(-\alpha x)\...
3
votes
2answers
108 views

Random complex eigenvalues and averages of traces

I have asked this in MSE here, but got no interesting answers. Suppose I have a random matrix $M$ of dimension $N$ which is real, but not symmetric. Suppose I know that, for large $N$, the marginal ...
3
votes
1answer
179 views

Inequality for exponential sum in Dvoretzky 1972

I'm currently trying to figure out the following inequality. It looks like an inequality for the exponential sum, but I can't verify it or find a source explaining it any further. Most likely it has ...
19
votes
1answer
1k views

How do mathematicians and physicists think of SL(2,R) acting on Gaussian functions?

Let $\mathcal{N}(\mu,\sigma^2)$ denote the Gaussian distribution on $\mathbb{R}$: $$ \mathcal{N}(\mu,\sigma^2)(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}.$$ A Gaussian ...
2
votes
0answers
48 views

Are sums extremal for subgaussian concentration?

Bobkov and Houdre https://projecteuclid.org/euclid.bj/1178291721 showed that among all $f:R^n\to R$ that are $1$-Lipschitz with respect to the $\ell_1$ metric, the variance is maximized by sums. ...
0
votes
1answer
91 views

Questions on a new definition of continuous multivariate distribution

For a univariate distribution or a univariate random variable, we call it continuous/absolutely continuous if its cumulative distribution function (CDF) is continuous/absolutely continuous. Now I am ...
3
votes
1answer
91 views

Approximating the mathematical expectation of the argmax of a Gaussian random vector

Let $X = \left( {{X_1},...,{X_n}} \right) \sim \mathcal{N}\left( {{\mathbf{\mu }},{\mathbf{\Sigma }}} \right)$ be a Gaussian random vector and $I = \mathop {\arg \max }\limits_{i = 1,n} {X_i}$. $I$ ...
1
vote
0answers
34 views

Stationary recursive sequence and nonzero probabilities

A while ago I posted the following problem: Suppose I have a two sided stationary sequence of random variables $\ldots,X_{-1},X_0,X_1,\ldots$ such that all finite dimensional joint densities $f(x_1,\...
2
votes
0answers
31 views

Probability of an n-sphere cap w.r.t. to the Angular Central Gaussian Distribution

Let $X \sim N[0,D]$ where $D$ is a diagonal matrix with entries, top-left to bottom-right, $d_1 \ge d_2 \ge d_3 \ge\dots\ge d_n > 0$. Let $e_1$ be the unit vector in $\mathbb R^n$, i.e. $[1,0,...,...