# 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.

7,283
questions

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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**

vote

**1**answer

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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votes

**2**answers

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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**3**answers

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 ...

**0**

votes

**1**answer

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**

vote

**1**answer

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....

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**1**answer

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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**0**answers

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**

votes

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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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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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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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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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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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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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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**

votes

**1**answer

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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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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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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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-...

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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)...

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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**

votes

**1**answer

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 $\...

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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]...

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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)...

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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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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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**1**answer

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**

votes

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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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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}_*\...

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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{...

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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?
$$f_{Y}(y)=\frac{10}{y\ln(10)\sqrt{2\pi}\sigma}\exp\left(-\frac{(-10\log_{10}(y) - \mu)^2}{2 \sigma^2}\right)$$

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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 ...

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**1**answer

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**

votes

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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 ...

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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 ...

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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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**1**answer

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, ...

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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

**2**answers

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 ...

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**1**answer

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 $\beta$ ...

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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 $(\Omega,\mathcal B(\Omega))$, ...

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**1**answer

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{...

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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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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 ...

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**1**answer

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:=\...

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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

**1**answer

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

**1**answer

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 $G:=X^\top(XX^\top + \lambda I_n)^{-1}X$. ...

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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**

votes

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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 {...