# 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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### 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
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1 answer
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### 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
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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: 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
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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
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1 vote
1 answer
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### 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 ...