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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18 views

Do we have the upper bound or the distribution of the following ratio?

Let $A=\{a_{ij}\}_{1\le i,j\le n}$ be an $n$ by $n$ normalized Gaussian random matrix with $E[a_{ij}]=0$ and $E[a_{ij}^2]=1/n$. Ordering its eigenvalues by $\lambda_1\le \lambda_2\le \cdots \lambda_n$ ...
2 votes
1 answer
225 views

An integral by rough path.

If $(b, \mathbb{b})\in \mathcal{D}^{\alpha}[0,T],\ \alpha\in (\frac{1}{3}, \frac{1}{2})$. $\mathcal{D}^{\alpha}[0,T]$ is the space of those rough paths $(b,\mathbb{b})$ such that $$ \|b\|_\alpha=...
0 votes
0 answers
62 views

Can we prove the following anti-concentration inequality of polynomials of square Gaussian variables?

Following this question Anti-concentration of Gaussian quadratic form. We have the following inequality: Let $X_1,\dots,X_n$ denote i.i.d. standard Gaussian random variables. For every $\epsilon>0$...
4 votes
4 answers
1k views

Is there an inequality relation between KL-divergence and $L_2$ norm?

According to the Pinsker inequality, we have the following inequality: \begin{equation} \delta_{TV} (p, q)^2 \leq \frac{1}{2} D_{KL}(p,q), \end{equation} where $\delta_{TV} (\cdot, \cdot)$ and $D_{KL}...
-1 votes
0 answers
19 views

Representation of characteristic function of Levy process

I'm studying levy processes and i just read this theorem. A few pages before this theorem, there is this one where is the measure with characteristic function: Now my question is, just with ...
4 votes
1 answer
476 views

Martingales and intersection of random walks

Let $G=(V,E)$ be a graph with $n$ vertices. Consider a pair of independent simple random walks $(X,Y)$ on the graph, each of length $L$ starting from a node $v \in V$. We denote a length-$L$ random ...
3 votes
1 answer
536 views
+100

How to get the lower bound of the following $\tau$?

Let $A=\{a_{ij}\}_{1\le i,j\le n}$ be an $n$ by $n$ normalized Gaussian random matrix with $E[a_{ij}]=0$ and $E[a_{ij}^2]=1/n$. Ordering its eigenvalues by $\lambda_1\le \lambda_2\le \cdots \lambda_n$ ...
0 votes
1 answer
67 views

Ito-Levy decomposition for $\alpha$-stable processes?

The Ito-Levy decomposition is well-known as a characterization of Levy processes. What does it give for the specific case of $\alpha$-stable Levy processes?
3 votes
1 answer
418 views

Equivalence of Itō and Stratonovich equations and how we ensure that the latter are well-defined

Remark: I've asked this question on MSE as well. Let $T>0$ $I:=[0,T]$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\in I}$ be a complete and right-continuous ...
-1 votes
1 answer
30 views

Second moment of stochastic integral wrt Levy Processes

I have a question about the second moment of the integral wrt Levy Processes. Let Z a Levy processe. We know that: And a few page later is written that by differentiation of the characteristic ...
3 votes
2 answers
164 views

Is every compound Poisson distribution a mixed Poisson distribution?

I asked and bountied this question at math SE but didn't get any answers, so I suspect that only experts (if anyone) may know the answer. The mixed Poisson distribution and compound Poisson ...
1 vote
1 answer
276 views

High-probability lower bound for norm of least squares solution when both design matrix $X$ and response vector $y$ are random (and independent)

Let $n,d \to \infty$ with $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ matrix independent rows uniformly distributed on the the unit-sphere in $\mathbb R^d$ and let $y$ be a ...
2 votes
0 answers
43 views

A version of Portmanteau theorem where $(\mu_n)_{n\in \mathbb N}$ is replaced by a net $(\mu_d)_{d\in D}$

Let $(E, d)$ be a metric space, $\mathcal C_b(E)$ the space of all real-valued bounded continuous functions on $E$, and $\mathcal P(E)$ the space of all Borel probability measures on $E$. For $f \in \...
2 votes
1 answer
417 views

Concentration of the norm of subGaussian random vectors

I will use the same notation and definitions in High Dimensional Probability, by Roman Vershynin. I have a sub-Gaussian vector $y$, in $\mathbb{R}^n$ and sub-Gaussian norm $C$ non dependent on $n$. I ...
5 votes
1 answer
179 views

Why does non-decreasing entropy imply actual convergence to that max entropy distribution?

Let $X_n$ be i.i.d with finite variance. Let $\bar X_n=\frac 1n \sum_{i=1}^nX_i$. It is a famous result that the continuous/differential entropy of the normalized average is non-decreasing. $$\mathrm ...
0 votes
1 answer
48 views

About Palm distribution

Can someone explain the Palm distribution? Or provide some information about Palm distribution. The article called 《A tutorial on Palm distributions for spatial point processes》 is hard to understand.
3 votes
1 answer
153 views

Is the limit of compound Poisson random variables a compound Poisson r.v.?

Let $Y$ be an infinitely divisible (I.D.) random variable. Let $\nu$ be any measure not necessarily finite: $\nu(\mathbb R)\leq \infty$. Suppose that $Y \sim (0, \nu,0)_0$ according to the notation on ...
3 votes
0 answers
79 views

Log of a truncated binomial

Let $X$ follow a binomial distribution with $n$ trials and success probability $p$, and let $0\leq k\leq n$. Are there any natural approximations or bounds for the ratio $$\frac{\boldsymbol{E}\log\...
25 votes
6 answers
5k views

Proof of Krylov-Bogoliubov theorem

Where can I find a proof (in English) of the Krylov-Bogoliubov theorem, which states if $X$ is a compact metric space and $T\colon X \to X$ is continuous, then there is a $T$-invariant Borel ...
1 vote
0 answers
64 views

Kleisli adjunction of the distribution monad

Let $\langle D , \mu, \eta \rangle$ be the distribution monad on $Set$ and let $Kl(D)$ be the Kleisli category on the distribution monad. I am interested in the adjunction between $Kl(D)$ and $Set$, ...
6 votes
0 answers
263 views

Mistakes in Logan and Shepp's famous paper on Young Tableaux?

In their landmark paper from 1977 named "A Variational Problem for Random Young Tableaux" Logan and Shepp obtained a number of results concerning asymptotic properties of Young Diagrams ...
2 votes
1 answer
68 views

Reference request: “A random integral and Orlicz spaces”

I need to find the following paper: “K. Urbanik and WA Woyczynski, A random integral and Orlicz spaces, Bulletin de l'Académie Polonaise des Sciences, Série des sciences mathématiques, astronomiques ...
10 votes
1 answer
1k views

Bounding the entropy of a convolution

Say we have a function $f:\mathbb{Z}_2^n \to \mathbb{R}$, such that $\sum _{x\in \mathbb{Z}_2^n} f(x)^2 = 1$ (so we can think of $\{ f(x)^2\} _{x\in \mathbb{Z}_2^n}$ as a distribution). It is natural ...
0 votes
0 answers
53 views

Kazamaki's condition

Consider the following problem from Etienne Pardoux book: Stochastic Differential Equations, Backward SDEs, Partial Differential Equations. It is about proving Kazamaki's condition. In part $(7)$ he ...
2 votes
1 answer
107 views
+50

Comparing diffusion processes in different metrics

I would like to know if it is possible to compare two diffusion processes defined on the same manifold $\mathcal{M}$ but with respect to different metrics say $g_1$ and $g_2$. Is there a way to apply ...
1 vote
1 answer
176 views

Hölder continuity of Radon transform of smooth function

Given an integrable function (e.g a probability density function) $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined by $$ R[f](w,b) := \int_{\mathbb R^n} \delta(x^\top w - b)f(x)...
1 vote
1 answer
66 views

Sub-exponential tail bound for Poisson multiplied by cosine of an independent uniform random variable

I am looking for the tail bound of the following random variable, hopefully of sub-exponential form: $\lambda_n^{-1}X_n\cos(\theta_n)$, where $X\sim Poisson(\lambda_n)$ with $\lambda_n\to 0$, and ...
2 votes
0 answers
80 views

How can I find the long-term summation of this random recursive sequence?

Suppose there is a stocastic random recursive sequence $\{b_n\}$, $n=1,2,...$, it evolves as follows: \begin{equation} b_i=\begin{cases} b_{i-1}+1, & \text{w.p. } \lambda, \\ 0, & \text{w.p. } ...
0 votes
0 answers
26 views

Concentration inequality of the $L^2$ norm of weighted vector with moment of eigenvalues of GOE

Let $A=\{a_{ij}\}_{1\le i,j\le n}$ be an $n$ by $n$ Gaussian random matrix with $E[a_{ij}]=0$ and $E[a_{ij}^2]=1/n$. Ordering its eigenvalues by $\lambda_1\le \lambda_2\le \cdots \lambda_n$ with ...
3 votes
0 answers
194 views

Riemannian metric induced by a stochastic differential equation

Following this paper, a diffusion process in $\mathcal{R}^d$ $$dX_t = f(X_t) \, dt + \sigma(X_t) \, dW_t ,$$ with $\sigma(x) \in \mathbb{R}^{d \times m}$ and $m$ dimensional Brownian motion can be ...
1 vote
0 answers
94 views

Wiener Integral and its distribution

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space. Let $(W(t))_{x \in \mathbb{R}^d}$ be a Gaussian random field. Then, we can define Wiener integral $\int_{\mathbb{R}^d} f(\xi) \, dW(\xi)$...
0 votes
0 answers
58 views

How are eigenvalues of two psd kernels related?

Suppose $K(x,x')$ and $R(y,y')$ are two positive semi-definite kernels on $(x,x')\in \mathbb {X}\times\mathbb{X}$ and $(y,y')\in\mathbb{Y}\times\mathbb{Y}$, respectively, and satisfying the following ...
4 votes
1 answer
198 views

Conditional expectation of linear combination of Rademacher RVs

Let $u, v \in \mathbb{S}^{d-1}$ be two unit vectors with $u \cdot v \geq c_1$. Let $Z \in \{-1, +1\}^d$ be a random sign vector where each coordinate is +1 or -1 independently with probability 1/2. I ...
1 vote
1 answer
70 views

Is there an inverse Lamperti transformation for diffusions?

The Lamperti transformation is commonly used to transform SDEs with state dependent coefficients into SDEs with constant diffusion. For multidimensional processes there are some conditions on the ...
3 votes
0 answers
158 views

Approximating any $d$-dimensional convex shape that occupies a constant fraction of its bounding box with a polytope having $\mathrm{poly}(d)$ facets

Given any convex set $A\in\mathbb{R}^d$, we denote by $V(A)$ its $d$-volume. Furthermore, given any two convex sets $A_1,A_2\in\mathbb{R}^d$, we denote by $V_{A_1,A_2}$ the $d$-volume of the symmetric ...
0 votes
1 answer
76 views

The ratio of spectral edge of the GOE matrix

Consider a $n\times n$ GOE random matrix. If we assume that $|\lambda_1|>|\lambda_2|\ge \dots \ge |\lambda_n|$, can we get the order of $|\lambda_1|/|\lambda_2|$ or even $\lambda_1/\lambda_2$? Any ...
3 votes
2 answers
521 views

Differentiability of characteristic functions and moments of the corresponding measure

Let $f$ be the characteristic function of a real-valued random variable $X$. It is known that if $f$ has a $k$-th order derivative (for some even $k$) then $\mu$ has a finite $k$-th order moment. Is ...
2 votes
0 answers
57 views

A variant of disintegration theorem where the assumptions on $f$ and $g$ are exchanged

I have recently read about about disintegration theorem, i.e., Disintegration theorem Let $X$ be a Polish space, $\mathcal X$ its Borel $\sigma$-algebra, and $\mu$ a Borel probability measure on $X$...
-4 votes
0 answers
36 views

Estimating a quantile from an unknown distribution

I have a variable $x \sim N(0,1)$, of dimension $p$. I want to find a scalar, $a$, such that $$\mathbb{P}\left(\frac{\sum_{i=1}^p (|x_i| - b_i) }{c} \geq a \right) = q, $$ where $c>0$ is a scalar, $...
5 votes
1 answer
288 views

Can an a.s. non constant continuous martingale be differentiable with nonzero probability?

Let $M$ be a continuous martingale such that almost surely, the sample paths of $M$ are not constant. Question: Is it true that $M$ is almost surely not differentiable?
1 vote
2 answers
123 views

On conditions for the existence of $h\in L^1$ such that $h>0$ a.e

I've been reading the chapter of Uniform Integrability on Probability Theory by Achim Klenke which has the following proposition. If $\mu$ is $\sigma$-finite, then there is a function $h \in L^{1}(\mu)...
1 vote
1 answer
228 views

What is this optimization problem called

Let $X$ be a set and $\mathcal{F}$ be a set of functions $f:X \to \Bbb{R}$ (for my purposes, it is fine to assume both sets are finite). For a probability distribution $\mu$ on $\mathcal{F}$, we ...
2 votes
0 answers
82 views

Optimal transport: how is the use of disintegration theorem valid in this construction of $\widetilde{\phi}$?

Let $X,Y$ be Polish spaces and $\mathcal P(X)$ the space of all Borel probability measures on $X$. Fix $\mu\in \mathcal P(X), \nu \in \mathcal P(Y)$. Let $\pi \in \Pi(\mu, \nu)$, i.e., $\pi \in \...
0 votes
1 answer
257 views

Hypothesis testing for not identically distributed random variables conditioned on the outcome of a subset

I encountered the following problem (I give more details of the problem at the end of the post) and I am trying to figure out the best way of performing a null hypothesis testing. I looked for similar ...
3 votes
1 answer
211 views

Quadratic variation of supremum of brownian motion

I would like to know if in some book or how could I compute the quadratic variation of the supremum of the bronian motion $S_t=\sup_{s\in[0,t]}W_s$ where $W$ is a Brownian motion. I was thinking ...
2 votes
2 answers
83 views

Concentration inequality for the spectral norm of the product of normalized Gaussian (and subgaussian) matrices in high dimensions

Let $X,Y$ be two $n\times n$ i.i.d. Gaussian matrices (entries are i.i.d N(0,1) and $X$ and $Y$ are independent). Consider their product normalized by the standard variance of entries $\frac{XY}{\sqrt ...
1 vote
1 answer
83 views

A question related to the CDFs of multivariate normal distribution

Let $\boldsymbol{\xi} = (\xi_1,\xi_2,\xi_3)$ such that $\xi_i\geq 0$and $\xi_1+\xi_2+\xi_3 = 1$. Let $Y\sim N_3(\boldsymbol{\mu}(\boldsymbol{\xi}), \mathrm{\Sigma}(\boldsymbol{\xi}))$, where the ...
2 votes
1 answer
113 views

Joint irreducibility and aperiodicity of two independent Markov chains

Let $(X_i)_i, (Y_i)_i$ be two independent Markov chains on Polish state spaces $\mathcal{X}, \mathcal{Y}$ and with kernels $P, Q$. Suppose that they are both $\psi$-irreducible and aperiodic and have ...
1 vote
0 answers
65 views

Stratonovich version of Girsanov

One version of Girsanov says that, that if $\mu_0$ is the law of a Brownian motion as a Borel measure on the space of continuous functions and we define the density $$\frac{d\mu}{d\mu_0}:=\exp\left(\...
3 votes
1 answer
71 views

Mutual information in large deviation theory

Many information theoretic quantities such as entropy and relative entropy appear in rate functions in large deviation theory (LDT). Is there any result in LDT that relates mutual information and rate ...

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