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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4
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2answers
132 views

Expected value of a ratio of squared normal and linear combination of squared normal [closed]

Given two positive constants $c_1,c_2$ and two independent standard normal random variables $a,b$, how to calculate the following expected value $$ \mathbb{E}\left[\frac{a^2}{c_1a^2+c_2b^2}\right] $$ ...
2
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1answer
165 views

Is $g(v)=\mathbb{E}[f(v+W)]$ a differentiable function of $v$ when $f$ is continuous and $W$ is multivariate normal?

Suppose $f$ is a continuous function on $\mathbb{R}^n$, and $W$ has a multivariate normal distribution on $\mathbb{R}^n$. If the expectation $$g(v)=\mathbb{E}[f(v+W)]$$ is defined for all $v \in \...
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1answer
217 views

Where should I submit a derivation of the CLT from the de Moivre-Laplace theorem?

By the central limit theorem (CLT), I mean the Lindeberg-Lévy CLT that says if $X_1,X_2,\ldots$ are i.i.d. random variables with $\mathbf{E}[X_1] = 0$ and $\mathbf{E}[X_1^2] = 1$, then $$ \frac{X_1+\...
2
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2answers
211 views

Convergence in probability of series of random variables

From the standard literature it is well known that for sequences of random variables $X_{1, n} \stackrel{P}{\rightarrow} X_1$ and $X_{2, n} \stackrel{P}{\rightarrow} X_2$ as $n \rightarrow \infty$ it ...
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47 views

"Cut norm" of conditional expectation has supremum on products of sets in sub-$\sigma$-algebra, or not?

I am reading Lovasz's book "Large networks and graph limits", and encountered the exercise that the stepping operator for graphons is contractive under the cut norm: $$||W_P||_\square\leq||W|...
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0answers
120 views

Are there any measurable spaces of functions

I am approaching this question from a probability perspective, and am hoping for some kind of framework to help understand all of this. I believe I may have even asked a similar question on here in ...
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0answers
52 views

Divergence between random variables after transformation

Let $X$ and $Y$ be random variables with laws $\mu_X$, $\mu_Y$ and $d$ be some $f$-divergence (e.g. KL, total variation, Hellinger). Writing $d(X,Y)$ for the divergence between $\mu_X$ and $\mu_Y$, ...
2
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1answer
225 views

Central limit theorem for weak correlated random variables

I have a sequence of weak correlated continuous random variables $\{X_i\}$ with bounded variance and $\operatorname{Cov}(X_i,X_j)\rightarrow0$ for $|i-j|\rightarrow\infty$. I was able to find a ...
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0answers
30 views

Uniform convergence of random Fourier quadratic forms

Let $\xi_k$, $k \in \mathbb {Z}$, be a sequence of independent Rademacher random variables. Is there a characterization of those families $a_{k,\ell}$, $k, \ell \in \mathbb {Z}$, of complex numbers ...
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1answer
117 views

How is close set condition used in this proof

I am reading Mathematical Statistics by ShaoJun (p. 56, Theorem 1.9). Does anyone know how "$C$ is closed" enters this proof? I don't know why the closedness condition is needed in this ...
1
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1answer
160 views

Random walks on Galton–Watson trees

I am working on a paper of Elie Aidekon : ‘Speed of the biased random walk on a Galton–Watson tree’ and have a question about one transformation in a proof: \begin{align} & 1+\frac{1}{1-\lambda}+\...
2
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1answer
170 views

Small ball Gaussian probabilities with moving center

I would like to prove (if possible, otherwise find a counterexample for) the following lemma: Let $(X,\|\cdot \|_X)$ be a separable Banach space. Additionally, we have a centred Gaussian measure $\mu$ ...
1
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1answer
207 views

An approximation problem w.r.t marginal distribution of coordinates of uniform random vector on high-dimensional unit-sphere

Let $X=(X_1,\ldots,X_d)$ be uniformly distributed on the sphere of radius $\sqrt{d}$ in $\mathbb R^d$. Fix a "sufficiently integrable" function $h:\mathbb R \to \mathbb R$, and define ...
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0answers
164 views

Show that the support restriction is not binding

Let $\mathcal{F}$ be the family of continuous distribution functions in $\mathbb{R}^3$ whose marginals are symmetric around zero and identical. Fix a vector of reals $\theta\equiv (\theta_1, \theta_2)\...
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0answers
85 views

Is this conjecture about the binomial and beta distributions true?

Let $X$ follow a binomial distribution with parameters $n$ and $p$, and also fix $k$ such that $1<k<n$. Define $$a = \mathbb{E}(X-k)^+$$ and $$b = \mathbb{E}\log\binom{X}{(X-k)^+}$$ where the ...
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0answers
56 views

Recurrence relation for the moments of the GOE

The Harer-Zagier formula provides a three term recurrence relation for the expected value of the single-trace operator $\mathrm {Tr}(X^k)$ where $X$ is a $N\times N$ matrix from the GUE. Is there an ...
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1answer
66 views

Compute the limit of trace of inverse of square of rank-1 perturbation of Wishart matrix

Let $a \ge 0$, $b,c>0$ be fixed constants, and let $X$ be an $m \times d$ random matrix with entries drawn iid from $N(0,1/d)$. Consider the random psd matrix $S := a 1_m 1_m^\top + b XX^\top + c ...
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0answers
36 views

Is there a local limit theorem for functions of Gaussian random vectors?

Assume that $\sqrt{n} (\boldsymbol{Z}_n - \boldsymbol{\mu}) \stackrel{\mathcal{D}}{\longrightarrow} \mathcal{N}(\boldsymbol{0},\Sigma)$, as $n\to \infty$, for some $\boldsymbol{\mu}\in \mathbb{R}^d$ ...
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1answer
110 views

Variant on Janson-type inequality

Let us suppose that we are in the setting of Janson's inequality for Poisson-type deviations of increasing events. Specifically, we have independent Bernoulli variables $X_1, \dots, X_n$, and events $...
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29 views

Fokker-Planck equation for reflected diffusion with time-varying reflecting boundaries

Suppose $X$ is a 1-d reflected diffusion on a time-varying domain $[l(t), r(t)] \subset (-\infty, \infty)$: $$dX(t) = b(t,X(t))dt + \sigma(t, X(t))dW(t) + dL_1(t) - dL_2(t),\ \ t\ge s \ge 0,$$ where $...
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1answer
160 views

From probability distribution in $\mathbb{R}^3$ to probability distribution in $\mathbb{R}^4$

I am working on a research paper where I need to investigate conditions for the existence of probability distributions satisfying certain characteristics. I have already asked a related question (here)...
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1answer
117 views

Trying to prove an inequality (looks similar to entropy)

I'm trying to prove the following inequality (or something similar, up to a constant factor in either side of the inequality): $$k\cdot\sum_{i=1}^{k}x_{i}\cdot\ln\left(x_{i}\right)\geq\sum_{i=1}^{k}x_{...
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0answers
78 views

Discrete-time model for spread of information when the probability of information transfer between each pair is known

[This question is cross-posted from MSE.] I'm interested in the behaviour of the following sort of system. We are given: a finite set $X$, a subset $A_0 \subset X$, and a function $f : X \times X \to [...
1
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1answer
115 views

Asymptotic limit of trace of random matrix $(aI_m + WW^\top)^{-1}$, where $W$ has iid rows from $N(0,\Sigma)$

Let $m$ and $d$ be positive integers with $m,d \to \infty$ such that $m/d \to \rho \in (0,\infty)$. Let $W$ be a random $m \times d$ matrix with iid rows $w_1,\ldots,w_m \sim N(0,\Sigma)$ for a ...
3
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1answer
117 views

Reference on variance of maximum of Gaussians

I have been told that if $(X_1, \ldots, X_n)$ is a centered Gaussian vector, then $$ \mathrm {Var} (\max_{1 \leq k \leq n} X_k ) \leq \max_{1 \leq k \leq n} \mathrm {Var} (X_k) . $$ What is a ...
4
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2answers
232 views

Average value of $\frac{x'A^2x}{x'A^3x}$ over surface of $n$-dimensional sphere

Suppose $A$ is a diagonal matrix with eigenvalues $1,\frac{1}{2},\frac{1}{3},\ldots,\frac{1}{n}$ and $x$ is drawn from standard Gaussian in $n$ dimensions. Define $z_n$ as follows $$z_n=E_{x\sim \...
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0answers
24 views

Homogeneity of coupling of 0-1 stochastic process

We use $2^n$ to denote the set of binary strings of length $n$; $2^{\leq n}$ to denote binary strings of length smaller than or equal to $n$; $\emptyset$ to denote empty string; $|\rho|$ to denote ...
3
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1answer
64 views

Donsker class and law of the iterated logarithm

Let $P$ be a probability measure on a measurable space $(E, \mathcal {E})$, and let $\mathcal {F}$ be a countable collection of measurable functions $f : E \to \mathbb {R}$ which is a Donsker class ...
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0answers
55 views

Is there any Azuma's inequality of continuous version?

For example, let $X_t$ be a martingale, $t \in [0,+\infty)$ satisfies $|X_t-X_0|\leq \int_0^t{c(s)ds}$, $c(s)\in [0,+\infty)$. Can we bound $P[|X_t-X_0|\geq \lambda], \forall \lambda \in R^+$ ?
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1answer
177 views

When is every Levy martingale of a process a continuous martingale?

Let $X_t$ be a real valued stochastic process, and $\mathcal H_t$ the the natural filtration of $X_t$. Under what conditions on $X$ does the following statement hold? For every $\mathcal H_\infty$-...
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0answers
76 views

Reference request: Introduction to stochastic control theory

I’m looking for a nice readable introductory text to stochastic control theory. Background wise, I know some general stochastic analysis and deterministic optimal control theory. Some criterion I’m ...
2
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0answers
137 views

Convergence of conditioned stochastic integral

Let $B_t$ be a standard Brownian motion, $f: [0, T] \to \mathbb R$ a bounded Borel measurable function, and $X_t$ a process independent of $B_t$ with sample paths that almost surely start at $0$, and ...
4
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1answer
119 views

Concentration of $k$-th pairwise distance of random points in a unit square

For $1\leq i \leq n$, let $X_i\sim \text{Uniform}(0,1)$, $Y_i \sim \text{Uniform}(0,1)$ be $n$ points chosen uniformly in the unit square. Denote the $k$-th smallest pairwise distances across the $n$ ...
0
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1answer
60 views

Bounding parameter satisfying a collection of inequalities

I have a set of equations with some inequality constraints that I expect generally does not have a unique solution. The equations take the form below: $$\alpha/N+(1-\alpha)x_1=a_1$$ $$\alpha/N+(1-\...
3
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0answers
162 views

Probabilistic behavior of greedy point selection in the plane

Let $\mathcal{X} = X_1,\dots,X_n$ be a collection of independent, uniform samples in the unit square. Let $\mathcal{S}=\{X_1\}$, and consider the following process: for $i=2,\dots,n$, let $x^*$ be ...
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1answer
173 views

Probabilistic interpretation of derivative of a Dirac delta function

Consider $g : \mathbb{R}^d \mapsto \mathbb{R}$ defines some surface $\Sigma$ in $\mathbb{R}^d$. Then I can define a random variable $X_1$ with support only on $\Sigma$ by using a pdf of the form $$p_1(...
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0answers
26 views

Hitting time of integrated/summed process and law of final jump $V_{T}$

Consider an integrated process $S_{t}=\int_{0}^{t} f(V_{s})ds$ for some Ito-diffusion $V_{s}$, a $C^{1}$ strictly positive function $f$ and a hitting time $T_{a}$ for the integrated process $S_{t}$. ...
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0answers
86 views

The probability of being able to solve a randomly generated maze

Let's say you have an $n \times n$ grid of cells, and a probability $p$ of having a wall between two adjacent cells, so you have a random square maze. What is the probability you can complete the maze ...
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0answers
38 views

Estimate of cumulative probability of geometric Brownian motion

Let $B_\tau$ be the standard BM, $t$ be the initial time, $s$ be the time variable, $r$ and $\theta$ are positive constants. We also assume that $x$ is the initial position of the below geometric ...
1
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1answer
121 views

Invariant distributions for iterated random variables (stochastic dynamical systems)

This is related to discrete dynamical systems, with the initial condition $X_1$ being a random variable with a non singular distribution. The system is driven by the iteration $X_{n+1} = g(X_n)$ for ...
3
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1answer
85 views

Best approximation of normal with $m$ atoms in Kolmogorov-Smirnov distance

Let $d_{KS}(F,G)= \sup_{x} |F(x) -G(x)|$ be the Kolmogorov-Smirnov distance between two cdfs $F$ and $G$. Question: Let $F_m$ be a cdf of distribution with $m$ atoms and let $\Phi$ bet the ...
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1answer
65 views

Empirical degree distribution of random $n$ vertices labeled rooted tree converges to Poisson distribution

I am reading Louigi's lecture note on random trees and graphs here. I get stuck on part (b), Exercise 1.2.3 on page 19, which says the following: Let $T_n$ be uniformly drawn from $\mathcal{T}_n$, ...
2
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0answers
60 views

Probability of filling a small ball before exiting a big one for $d=2$

Let $S_n$ be the simple random walk in dimension $d=2$. Let $0<r<R$ and $\alpha \in (0,1)$. Let $B_r$ denote the $\{x \in \mathbb Z^2: \|x\|\le r\}$ where $\|\cdot\|$ is the Euclidean norm. ...
5
votes
2answers
452 views

Distribution of some sums modulo p

Fix a finite set of integers $S$ and a prime number $p$. Let $(a_1, a_2, \dotsc, a_n)$, $(b_1, b_2, b_3, \dotsc, b_n)$ be two sequences of integers where the numbers $a_i$ and $b_i$ are chosen ...
8
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1answer
332 views

Question about estimating random symmetric sums modulo p

Let $n > 0$ be a positive integer (large) and $p > 2$ a fixed prime number. What is the probability that $$\sum_{ 1 \leq i < j \leq n} a_ia_j = 0 \mod p$$ where $a_1, a_2, \dots a_n$ are ...
0
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0answers
55 views

Empirical estimation of Brenier map from data

Let $f:\mathbb R^d \to \mathbb R$ be a "nice" (say, continuous) function define $A = A_f := \{x \in \mathbb R^d \mid f(x) \ge 0\}$ and $B =B_f:= \{x \in \mathbb R^d \mid f(x) \le 0\}$, and ...
0
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2answers
177 views

Conditions for existence of a distribution with full support

Consider a $6\times 1$ continuous random vector $$ \eta\equiv (\eta_1,\eta_2,..., \eta_6) $$ satisfying the following property: $$ \underbrace{\begin{pmatrix} \eta_1\\ \eta_2\\ \eta_3 \end{pmatrix}}_{\...
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0answers
48 views

Regularity of Gaussian process sample paths

Consider a Gaussian process on $[0,1]$ given by a kernel function $K: [0,1]^2\to\mathbb{R}$. Under what conditions can we conclude that the sample paths are $C^k$ with probability 1? This question is ...
6
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1answer
394 views

Graphs resembling the math genealogy graph must have concentration in a small number of families?

I was talking with a non-mathematician the other week at a workshop about the fact that many mathematicians, like myself, are indexed in the math genealogy database. We talked a little about how many ...
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0answers
69 views

Longest close common subsequence but for continuous random variables

We have two copied sequences of correlated continuous positive random variables that are independent of each other $(X_{n})\perp(Y_{n})$ and equal in distribution $X_{n}\stackrel{dis}{=}Y_{n}$ for ...

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