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
votes
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 ...
1
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,...,...
