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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6
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2answers
228 views

Can Birkhoff's ergodic theorem for integrable functions easily be deduced from Birkhoff's ergodic theorem for bounded functions?

It seems to me that a considerably simpler proof [see below] of Birkhoff's ergodic theorem can be obtained for bounded observables than for more general $L^1$ observables. Therefore, I feel like it ...
1
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1answer
71 views

Estimating the average of two gaussians' mean with minimal squared error

This is a follow-up to my previous question. Assume that $X\sim \mathcal N(\mu_1,\sigma_1^2)$ and $Y\sim \mathcal N(\mu_2,\sigma_2^2)$. I want to estimate $\frac{\mu_1+\mu_2}{2}$ after observing $X,Y$....
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2answers
73 views

What is the limiting marginal distribution of a fixed number of coordinates of a random point drawn uniformly on large-dimensional sphere?

Let $X=(X_1,\ldots,X_d)$ be uniformly-distributed on the sphere of radius $\sqrt{d}$ in $\mathbb R^d$. It is well-known that in the limit $d \to \infty$, the marginal distribution of $X_1$ converges ...
2
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3answers
206 views

Dominated convergence theorem when the measure space also varies with $n$

Let $(f_n)_n:X \to \mathbb R$ be a sequence of measurable functions on a measurable space $X$ converging pointwise to a function $f:X \to \mathbb R$, and let $(\mu_n)_n$ be a sequence of finite ...
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1answer
47 views

Sufficient conditions for decomposition of a bounded random variable into several small pieces

Given a random variable $X$ with $\mathsf{supp}\, X \subseteq [0,1]$ and $n$ positive numbers $h_1,\cdots,h_n$ with $\sum_{i=1}^n h_i=1$, I want to know some sufficient conditions for decomposing $X$ ...
1
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1answer
41 views

tail probability of max of Gaussians

I'm trying to follow an argument in C. Giraud's "High Dimensional Statistics" (2nd Ed, p. 11 / $\S$ 1.2.3). The specific page is accessible via Google Books here but the formatting is awful....
1
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1answer
52 views

Estimating the average of two gaussians' mean

Assume that $X\sim \mathcal N(\sigma_1,\mu_1)$ and $Y\sim \mathcal N(\sigma_2,\mu_2)$. I want to estimate $\frac{\mu_1+\mu_2}{2}$ after observing $X,Y$. In my setting, $\sigma_1,\sigma_2$ are known ...
3
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0answers
85 views

Bounds on the entropy of the 2D Ising model

I am interested in good estimators of (or analytical bounds on) the entropy $\mathsf{H}_\beta:=-\sum_{\mathbf{x}} P(\mathbf{x})\log_2(P(\mathbf{x}))$ of the two-dimensional Ising model (with no ...
2
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2answers
72 views

Does fixed allocation increase the proportion of positively drifted Brownian motions surviving forever?

This is a continuation of Number of drifted Brownian motions that never hit zero under allocation For each $n\ge 1$, consider $X^i_t=1+\beta t + W^i_t$ for $i=1,\ldots n$ and $t\ge 0$, where $\beta>...
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0answers
20 views

On rank distribution of certain matrix products

We are in $\mathbb F_2$ and we have a rank $k$ matrix in $\mathbb F_2^{n\times n}$ where $k\in\{1,\dots,n\}$ and we fix $j\in\{1,\dots,n-1\}$. We pick $n-j$ uniformly random matrices $M_1,\dots,M_{n-j}...
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2answers
75 views

Number of drifted Brownian motions that never hit zero under allocation

For each $n\ge 1$, consider $X^i_t=1-\beta t + W^i_t$ for $i=1,\ldots n$ and $t\ge 0$, where $\beta>0$ and $(W^i_t)_{t\ge 0}$ are independent Brownian motions. $\phi\equiv \big((\phi^1_t)_{t\ge 0},\...
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2answers
74 views

Exponential decay of Fisher information along the OU semigroup

I read from a paper that there is a "well-known" exponential decay of Fisher information along the OU semigroup, that is $$J(\nu^t\mid\gamma)\leq e^{-2t}J(\nu\mid\gamma),$$ where $\gamma$ is ...
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0answers
33 views

Canonical representation of the a probability distribution for Hammersley Clifford Theorem

I'm reading the following paper http://www2.stat.duke.edu/~scs/Courses/Stat376/Papers/GibbsFieldEst/BesagJRSSB1974.pdf On page 7 they give the result that $$Q(\textbf{x}) = \sum_{1 \leq i \leq n} ...
4
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0answers
51 views

Does minimum mean-square error characterize distribution?

Let $X$ be a real random variable, and, for each $t \geq 0$, let $X_t = X + \sqrt {t}Y$ where $Y$ is a standard normal independent of $X$. The quantity $$ R(t) = E \big ( [ X - E (X |X_t)]^2 \big),...
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0answers
69 views

Space derivative of Brownian local time

Let $\{L_x(t):(t,x)\in [0,T]\times\mathbb R\}$ be the local time of a Brownian motion $(B_t)_{t\in [0,T]}$, I know that the map $x\mapsto L_x(t)$ is $\alpha$-Hölder for $\alpha<1/2$ (uniformly in ...
0
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1answer
117 views

Components of a gram matrix and its eigenvalues

The Gram Matrix is defined as $$\sum_{i=1}^n X_iX_i^T$$, where $X_i$ is drawn from the unit sphere based according to some continuous distribution (Relation between eigenvalues and the gram matrix for ...
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2answers
88 views

Convergence of stationary distributions of a sequence of Markov Chains

I fairly new in the field of Stochastic Processes and Markov Chains so excuse my ignorance. My question is: If we have a sequence of Markov chains such that each one has a stationary distribution $\pi^...
6
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3answers
255 views

Curvature function as a random variable with uniform distribution

Let $(S,g)$ be a Riemannian surface. Then the curvature function $\kappa: S\to \mathbb{R}$ can be counted as a random variable. So it produces a probability density function $f_g:\mathbb{R}\to \...
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0answers
75 views

Tail bounds for random Gaussian chaos?

Let $g = (g_1, \dots, g_d)$ be a sequence of independent standard Normal random variables, and suppose $\Sigma$ is a $d \times d$ (deterministic), real, symmetric, positive definite matrix. The Hanson-...
1
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1answer
69 views

Estimates on the discrepancy of random sequences

The discrepancy of a $[0,1]$-valued sequence $n \mapsto \alpha_n$ is the quantity $$D(N; \alpha) \stackrel{\text{def}}{=} \sup_{(a,b) \subset [0,1]} \left|\frac{\#\{1 \leq n \leq N : \alpha_n \in (a,b)...
5
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3answers
452 views

A toy model of heat death

Motivation: This is a toy model of how a closed system will always evolve towards the distribution of maximal entropy, where no further transfer of heat/energy is possible. Problem set up: Fix a ...
4
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1answer
121 views

Conditions for the SDE be transitive

This question was previously posted on MSE. Let $f:\mathbb R^3 \to \mathbb R^3$ be a smooth Lipschitz function (bounded if needed), and $W_t$ a $3$-dimentional Brownian motion. Consider the SDE on $\...
2
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0answers
83 views

Continuity of the entropy of the solution of a parabolic PDE at $t=0$

Consider the following initial value problem for a parabolic PDE : $$\begin{cases} \textrm{div}\big(A\,\nabla u(t,x) + b(x)\, u(t,x)\big) \,=\,\partial_t u(t,x) \quad x\in\mathbb R^d\,,\ t>0 \\[4pt]...
3
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1answer
242 views

A quantity associated to a probability measure space

Let $(S,P)$ be a (finite) probability space. We associate to $(S,P)$ a quantity $n(S,P)$ as follows: The probability of two randomly chosen events $A,B\subset S$ being independent is denoted by $n(S,P)...
0
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1answer
68 views

An inequality for a "generalised random energy model"

Let, for all $i, j$, $Z_{i,j}$ be a standard normal, chosen iid. For each $n\geq 1, k\geq2$, define the Hamiltonian $H_{n,k}: [k]^n \to \mathbb{R}$ by $$(j_1,j_2,\ldots,j_n) \mapsto \sum_{i=1}^n Z_{i, ...
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0answers
37 views

infinitesimal generators for G/G/1 queue

I read the infinitesimal generator for the M/M/1 queue and thought to generalize to the G/G/1 queue. More specifically, though the queue length process is not Markovian anymore, we could consider an ...
2
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1answer
81 views

Necessary moment condition for the law of the iterated logarithm

Let $X_1, X_2,\ldots$ be iid random variables, and let $S_n = X_1 + \cdots + X_n$, $n \geq 1$. It seems classical that if $$ \limsup_{n \to \infty} \frac {|S_n|}{\sqrt {2 n \ln \ln n}} < \infty ...
2
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2answers
107 views

Integral of product of Hermite polynomials w.r.t marginal distribution of first two-coordinate of random vector on unit-sphere

This question is related to: https://math.stackexchange.com/q/4270522/168758 Let $H_n(x) \in \mathbb R[x]$ be the probabilist's $n$th Hermite polynomial. This an $n$th degree polynomial given by the ...
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0answers
76 views

cemetery tree $\delta$

Let $T_*$ be a tree on which the parent $e_*$ of the root is added and $x\in\mathbb{V}$ a vertex in the Ulam-Harris tree. Then the tree $T_*^{\leq x}$ is defined as \begin{cases} \mathsf{T}_*\...
0
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1answer
34 views

PDF of the summation of L lognormal RVs

Given the following summation $$\gamma = \sum_{l=1}^{L} y_{l},$$ where the PDF of $Y$ follows the lognormal distribution and is given by $$f_{Y}(y)=\frac{10}{y\ln(10)\sqrt{2\pi}\sigma}\exp\left(-\frac{...
0
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0answers
40 views

Lognormal PDF in terms of the Meijer-G function

Is it possible to write this lognormal PDF in terms of the Meijer-G function? $$f_{Y}(y)=\frac{10}{y\ln(10)\sqrt{2\pi}\sigma}\exp\left(-\frac{(-10\log_{10}(y) - \mu)^2}{2 \sigma^2}\right)$$
8
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2answers
741 views

Stochastic dominance between (products of) binomials

Suppose $p \leq q \leq 1/2$, and $n,m\geq 1$ two integers. Let $X\sim \mathrm{Bin}(n,p)$, $Y\sim \mathrm{Bin}(m,p)$ and $X'\sim \mathrm{Bin}(n,q)$, $Y'\sim \mathrm{Bin}(m,q)$ be independent. Is it ...
3
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1answer
109 views

Comparison of probabilities that drifted Brownian motion never hits barriers

Let $k , h: \mathbb R_+\to [0,1]$ be non-decreasing and right continuous s.t. $k(t)\le h(t)$ for all $t\ge 0$. Define $\tau_{k}$ (resp. $\tau_h$) by $$\tau_k : = \inf\{t\ge 0:2+\beta t+ W_t \le k(t)\}\...
4
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1answer
198 views

What is the number of finite Dynkin systems?

(This is a spin-off of Determine the minimal elements of a Dynkin system generated by a finite set of finite sets) Let $\Omega$ be a finite set. A Dynkin system on $\Omega$ is a subset of the power ...
2
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2answers
120 views

Conditions for the existence of von Neumann-Morgenstern utility on a Polish space

Let $X$ be a Polish space, i.e. a separable complete metric space. Any Borel probability measure on $X$ must be locally finite, outer regular and tight. Let $\mathcal{P}(X)$ be the set of all Borel ...
1
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1answer
136 views

Approximate the singular values of a certain random dot-product kernel matrix (in the sense of El Karoui, Cheng-Singer, etc.)

Let $g:\mathbb R \to \mathbb R $ be a continuous function which is "sufficiently smooth" (e.g $\mathcal C^3$) around $0$, and "sufficiently integrable" (e.g integrable w.r.t $N(0,...
2
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1answer
67 views

Galton-Watson process: branching property

I am looking for the definition of the ‚branching property‘ of a Galton Watson process. Can someone give me an example about it? It looks to me like an independence. I have a branching processes book, ...
3
votes
2answers
174 views

Bounded density for diffusions with diffusion coefficients bounded away from $0$

Consider a diffusion given by $$X_t=\int_0^t a(s,X_s)\,dW_s$$ for $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and $a$ is a smooth enough positive function bounded away from $...
5
votes
2answers
193 views

A comparison of diffusions

Consider two diffusions given by $$X_j(t)=\int_0^t a_j(s,X_j(s))\,dW_s$$ for $j=1,2$ and $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and the $a_j$'s are smooth enough ...
1
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1answer
238 views

Occupation times for two-state Markov processes

Consider a two-state Markov process in continuous time, with states labelled $A$ and $B$. The transition rates for going from state $A$ to $B$, and state $B$ to $A$ are $\alpha$ and $\beta$ ...
2
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0answers
31 views

Continuity of translation operator in fractional white noise analysis

Fix $H\in(\frac{1}{2},1)$, and let $\Omega:=C_0([0,T],\mathbb R^d)$ be the space of $\mathbb R^d$-valued continuous functions. There is a probability measure $P^H$ on $(\Omega,\mathcal B(\Omega))$, ...
3
votes
1answer
129 views

How to prove excursion process is a Poisson point process?

This question comes from book Ju-Yi Yen and Marc Yor P59 and P60, On page 59, "Define $\mathcal{Z}_\omega=\{t:B_t(\omega)=0\},$ and $\tau_l$ is the inverse local time. The complement of $\mathcal{...
2
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0answers
84 views

Equivalence between notions of dynamical coupling as defined by Villani in his book Optimal Transportation: Old and New

$\DeclareMathOperator\law{law}$In Villani's book he presents the following notions of dynamical couplings: Let $(X,d)$ be a Polish space. A dynamical transference plan $\Pi$ is a probability measure ...
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0answers
63 views

When does an RKHS contain another?

Consider a psd kernel function on the unit-sphere in $\mathbb R^d$ off the form $K(x,x') = \varphi(x^\top x')$ for some $\varphi:[-1,1] \to \mathbb R$, and let $\mathcal H_\varphi$ be the induced ...
1
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1answer
52 views

Random walks on GW-trees (regeneration epochs/survival set)

Let $\Gamma_0,\Gamma_1,...$ be regeneration epochs. If $(X_n)_{n \in \mathbb{N}}$ is a $\lambda$ biased random walk on a Galton-Watson tree, than the regeneration epochs are defined as: $\Gamma_0:=\...
4
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0answers
137 views

Determine the minimal elements of a Dynkin system generated by a finite set of finite sets

(This is a refined version of https://cs.stackexchange.com/q/144371) Let $\Omega$ be a finite set. A Dynkin system on $\Omega$ is a subset of the power set of $\Omega$ containing $\Omega$, which is ...
3
votes
1answer
139 views

Well-definedness of maximum likelihood estimation

Consider a family $\{\mu_\theta:\theta\in\Theta\}$ of probability measures on a measurable space $X$. Given $x\in X$, the maximum likelihood estimate is the value of $\theta$ which maximizes the ...
2
votes
1answer
66 views

Asymptotics of $w^\top G^2 w$, where $w$ is a unit-vector, $G:=X^T(XX^T+t I_n)^{-1}X$, $t > 0$, and $X$ is an $n\times d$ gaussian random matrix

Let $X$ be an random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $w$ be a unit-vector in $\mathbb R^d$. With $\lambda>0$, and define $G:=X^\top(XX^\top + \lambda I_n)^{-1}X$. ...
1
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0answers
30 views

Hitting distribution of a sub-lattice

Let $\{X_n\}$ be the simple random walk in dimension $d=2$. Consider the distribution $$ p^M(0,y)=\mathbb{P}_0[X_{\tau_{M}}=M \cdot y] $$ where $\tau_M:=\inf \{n \ge 1: X_t \in M \cdot \mathbb Z^d \}$ ...
3
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0answers
39 views

References on precise Large Deviations Principle/Laplace method for binomial sum

I am looking for an estimate of the following sum/expectation: \begin{align*}%$ J_n & = \mathbb{E}\left( e^{n f(X_n) + \log(n) g(X_n) + h(X_n)} \right) \\ & = \frac{1}{2^n} \sum_{k = 0}^n {...