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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21 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 $\lambda_1\le \lambda_2\le \cdots \lambda_n$ ...
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-1 votes
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20 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 ...
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-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 ...
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63 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 $\epsilon>0$...
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2 votes
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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 \...
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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.
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3 votes
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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\...
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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 $Set$, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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. } ...
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 ...
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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 ...
  • 138
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
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, $...
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 $X$...
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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
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 ...
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 \...
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4 votes
1 answer
199 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 ...
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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 ...
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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
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
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)...
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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
537 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 $\lambda_1\le \lambda_2\le \cdots \lambda_n$ ...
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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 ...
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-1 votes
0 answers
73 views

Krein-Rutman theorem

Perron-Frobenius theorem guarantees that the largest eigenvalue of an irreducible positive matrix is positive and the corresponding eigenvector has all entries positive. The generalization of this is ...
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-3 votes
0 answers
35 views

Why WGM can approximate Expectation?

In the image above, I don't understand equation (14), where $E_x[y(x)]$ is approximated to $WGM(y(x))$. (All the equations above seem to only explain how $WGM(y(x))$ is equal to $\exp(Ex[f(x)])$). Why ...
6 votes
1 answer
170 views

Origin of the term "connective constant"

Let $G$ be a vertex-transitive locally finite graph and $c_n$ the number of self-avoiding walks in $G$ starting from some fixed vertex $v_0$. One can easily see that $c_{m+n} \leq c_m c_n$ and hence ...
0 votes
1 answer
135 views

Vertex degree on random graphs

Let $p = d/n$ with $d$ constant. How do I prove that, with high probability, $G_{n,p}$ contains a vertex of degree at least $(\log n)^{1/2}$, where $G_{n,p}$ is a graph with $n$ vertices and the ...
0 votes
1 answer
53 views

Probability of accurate sparse recovery

Suppose $\mathbf{A}_{k\times n}$ ($k<n$) is a matrix whose entries are generated i.i.d. from Gaussian distribution and $\mathbf{s}_{n\times 1}$ is a sparse vector with $m$ sparsity (i.e., $\|\...
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1 vote
1 answer
133 views

A question about the proof of the Levy-Khintchine representation Theorem

I'm studying Infinitely Divisible random variables using this Lecture Notes. And I have a question that is driving me crazy. In the proof of the "only if" part of the Levy-Khintchine ...
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0 votes
0 answers
45 views

(In)dependence between spatially transformed stochastic processes

Consider the C-shaped region $C\subset \mathbb{R}^2$ shown below, which has arms $\text{I, II}$ $\subset C$ given by $\text{I}$ $= I_0\times I_1$ and $\text{II}$ $= I_0\times I_2$ for some intervals $...
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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 ...
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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 ...
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0 votes
0 answers
22 views

Finding the resolvent of two coupled random matrix system using supersymmetry

I'm trying to follow the result in this work. Let me briefly introduce the problem. I have a $2N\times 2N$ matrix $$ H=\begin{pmatrix}H_1 & V \\ V^{\dagger} & H_2\end{pmatrix}. $$ Here both $...
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2 votes
0 answers
60 views

General ballot theorem: sum of independent but not identically distributed random variables?

Is there ANY ballot-type result for random walk $S_n:=\sum_{i\le n}X_i$ that allows for independent but not identically distributed random variables $X_i$, up to some uniform concentration conditions ...
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0 votes
0 answers
79 views

Gauss maps and stopping times

Given $x \in (0,1)$, the Gauss map is defined by $G(x)=x^{-1}-[x^{-1}]$ with an invariant measure $d\mu(x)=(1+x)^{-1}dx$ on $(0,1)$. Let $R \in (0,1)$, define $N_R(x)=\inf \{n\ge 1 : \prod_{i=0}^{n-1}...
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0 votes
0 answers
22 views

Size of the second largest strongly connected component in random directed graphs with fixed in- and out-degree sequences

It has been known for long (Molloy and Reed 1995) that in a supercritical undirected configuration model, that is when $E[D(D-2)]>0$, $D$ degree of a uniform vertex, the size of the second largest ...
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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 ...
-1 votes
0 answers
71 views

Markov chain + variable

Consider $P(X,Y)$ discrete and a Markov chain $Y_0 \rightarrow Y_1 \rightarrow \dotsc \rightarrow Y_{L-1} \rightarrow Y_L$ with $Y_0 := Y$. The chain $Y_L \rightarrow Y_{L-1} \rightarrow \dotsc \...
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4 votes
1 answer
123 views

Decreasing tail integrals for nonnegative random variable $X$

Let $X$ be a nonnegative random variable with density function $f(x)$, distribution function $F(x)$, survival function $S(x)=1-F(x)$ and finite first and second moments. Let also $$\ell(x):=\frac{1}{...
7 votes
0 answers
136 views

Approximating any convex shape in $\mathbb{R}^d$ with a polytope having $\mathrm{poly}(d)$ facets

We denote by $V(A)$ the $d$-volume of any convex set $A$. Furthermore, given any two convex sets $A,B\in\mathbb{R}^d$, we denote by $V_{A,B}$ the $d$-volume of the symmetric difference $V\left(A \...
3 votes
1 answer
66 views

Quasi-random vs pseudo-random graphs

My question is somehow concerning terminology on extremal graph theory. Is there any difference concerning the notion of quasi-random graph and the notion of pseudo-random graph? My feeling is that ...
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4 votes
2 answers
169 views

Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation

I'm reading a proof of below theorem from this paper. Theorem A.3. Let $\Omega$ be a locally compact normal Hausdorff space. Let $\left\{\mu_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ and ...
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9 votes
2 answers
404 views

Sum of a Poisson point process

Let $Z_1, Z_2, \dots$ be a Poisson point process on $[0, 1]$ with intensity function $1/z$. What is the distribution of the sum $Z = \sum_{i=1}^\infty Z_i$? One can construct $Z_1, Z_2, \dots$ by ...

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