# 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.

7,994
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### Interpolation theorem for general rough paths

In Friz and Hairer's notes on rough paths, there is exercise 2.9 which is called the "interpolation theorem". It says that if you have a sequence of rough paths $\mathbf X^n=(X^n,\mathbb X^n)...

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### How is $\mathbb E[ \int_0^T H_s^2 \mathrm d s] < \infty$ important for this claim?

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space. Let $H=(H_t, t\ge 0)$ be a stochastic process with continuous trajectories. Fix $T>0$. For $n \ge 1$, we define
$$
H_{s,n} := \sum_{i=1}...

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50
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### Stability of SDE fBM

Consider an n-dimensional Ito process
$$
X_t^x = x + \int_0^t\, \alpha(s)ds + \int_0^t\,\beta(s)\,dB^H(s),
$$
where $1/3<H<1$ is the Hurst parameter for an $n$-dimensional fractional Brownian ...

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85
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### Maximal mutual information between a continuous and a discrete random variables

Let $X\sim \mathcal{N}(\mu,\sigma^2)$ be a Gaussian random variable with random mean $\mu\sim {\sf Bernoulli}(p)$, i.e., $\mu=1$ with probability $p$ and $\mu=0$ with probability $1-p$. In other words,...

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### Precise asymptotics for moments of order statistics of normal distribution

Let $X_1, \cdots, X_n \sim N(0,1)$ be i.i.d. normal random variates. I am interested in understanding the first two moments of the quasi-range $X_{(n)}-X_{(n-1)}$ (i.e., the maximum value minus the ...

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### Connectivity constant for lattices

A celebrated result due to Duminil-Copin and Smirnov states that the connectivity constant for the honeycomb lattice is equal to $\sqrt{2+\sqrt{2}}$.
My question is the following: apart from the ...

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43
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### Lévy measure and jump behaviour of the corresponding Lévy process

Let $(X_t)_{t \ge 0}$ be a Lévy process on $\mathbb R$ with Lévy measure $\nu$.
Define the jump counting measure $N(t, A) = \lvert\{s \in [0, t] \mathrel: \Delta X_s \in A\}\rvert$
where $\Delta X_s$ ...

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### When is the image of $T \colon \ell^2 \to \ell^2$ a Gaussian random variable?

In finite dimensions, if $T$ is a linear operator and $x$ is a (centered) Gaussian random variable, then $Tx$ is again a (centered) Gaussian random variable.
Now suppose that $x$ is a (say, centered) ...

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### A question about an intepretation of certain probability

Consider a polynomial $p(z)= \sum_0^n a_i z^i.$ In the literature there are numerous bounds about the roots of $p(z)$.Now once we prescribe certain dsitribution to the coefficients ,the bound itself ...

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### Measurability of a process defined by an integral

Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$ Let $\mathcal{P} \subset \mathcal{B}(\mathbb{R}_+) \otimes \mathcal{F}$ the $\sigma$-algebra generated by $\{\{0\}\times F_0\}...

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### Does the following expectation-based inequality hold?

Let $\mathcal{F}$ be the space of all functions that uniformly and independently map the alphabet $\mathcal{X}$ to the set $\{1,2,\ldots,A\}$. Let $p(x|y)$ be an arbitrary conditional probability ...

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### Probability bounds of some ranked version of Dirichlet distribution

Recently I have come across a distribution defined on the open ranked simplex $\nabla^{n-1}_+ = \{\vec x \in \mathbb{R}^n:\sum_{k=1}^n x_k =1, x_1 \geq x_2 \geq \cdots \geq x_n > 0\}$, whose ...

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### Order of orthant probabilities in a prolate multinormal distribution

This is inspired by the negative answer to the conjecture in Which orthant probabilities are the largest? (For a multivariate normal distribution).
Suppose $X$ has the $k$-dimensional multivariate ...

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### Is svd of a Gaussian iid matrix corresponds to Haar measure on the Stiefel manifold?

I want to draw a matrix $A\in \mathbb{R}^{n\times k}$ uniformally at random from the Stiefel manifold $\mathbb{V}_k(\mathbb{R}^n)$, that is from the collection of all $n\times k$ matrices $A$ such ...

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### Predictability of the mild solution of a SPDE

Consider the following theorem (picture below) taken from Pardoux's lecture notes: Stochastic partial differential equations available at scholar google: https://scholar.google.ca/scholar?q=etienne+...

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### Popular mistakes in probability

$\DeclareMathOperator\Var{Var}\DeclareMathOperator\Bern{Bern}\DeclareMathOperator\Pois{Pois}$Question: What not-trivial mistakes do students often make when solving problems in probability theory, ...

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### Under which conditions Mean Square Continuity implies Sample Continuity for Gaussian Processes?

First, let us give the setting.
Let $(\Omega, \Sigma, \mathbf{P})$ be a probability space, let $T$ be some interval of time, and let $X: T \times \Omega \rightarrow S$ be a stochastic process.
By Mean ...

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### Vector space of random variables [closed]

Let $(\Omega, \mathfrak{S}, \mu)$ be a probability space and let $R(\Omega)$ be the space of all real-valued random variables $X: \Omega \to \mathbb{R}$ w.r.t. the above probability space having ...

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+50

### Shift-ergodic stochastic processes in continuous time

Let $\mathscr{C}:=\{\gamma : \mathbb{R}_+\rightarrow\mathbb{R}^n \mid \gamma \ \text{ continuous}\}$ be the set of all $\mathbb{R}^n$-valued paths over $[0,\infty)$. Endow $\mathscr{C}$ with the $\...

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### Riemannian submanifolds of $2$-Wasserstein space

In the article "Wasserstein Geometry Of Gaussian Measures" by Asuka Takatsu the author shows how the space of d-dimensional Gaussian probability measures with non-singular covariance ...

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### Measurability in a product space of a set defined only along its fibers

Consider the probability space $([0,1],\mathcal{B}([0,1]),\lambda)$, where $\mathcal{B}([0,1])$ denotes the Borel $\sigma$-algebra in $[0,1]$ and $\lambda$ is the Lebesgue measure in $[0,1]$. Then, ...

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### The Rate of converging to the stationary distribution given a time in-homogenous but fast converging transition probabilities

Let $P_n$ be a sequence of transition probabilities and $X_n$ be the corresponding Markov chain. That is , $X_n=d_0P_1...P_{n}$, where $d_0$ is the initial distribution. Suppose each $P_n$ has its ...

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### Asymptotic properties of weighted random walks / infinite convolutions of random variables

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d. real-random variables. Let further $0<c<1$. I'm interested in the asymptotic properties of
$$
\sum_{k=1}^n c^k X_k.
$$
I can prove that this ...

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### Lipschitz function of a sub Gaussian vector

I have been struggling with the following question.
Let $X \in \mathbb{R}^n$ be a $K$ sub Gaussian random vector (i.e. $\|\langle u, X \rangle\|_{\psi_2} \leq K$ for all $\|u\|=1$) and let $f : \...

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### Is the unconditional variance of a RV an upper bound for the variance of any conditional expectation of the RV?

Let $X$ and $Y$ be continuous random variables with finite first and second moments. Then, is it true that $Var[X]\geq Var[E(X|Y)]$?

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### What is the probability that after throwing n balls into k bins uniformly we will have a different number of balls in each bin?

I want to know if it is possible to compute the following problem (Or at least give an estimation on the lower bound) :
Given $n$ balls and $k$ bins where $n>>k$, we throw $n$ balls into those $...

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### Fluctuation-dissipation theorem for Markov processes

In the context of particle systems of non-gradient types (see e.g. here, Step 2 on page 633), I recently encountered the concept of fluctuation-dissipation theorem (FDT). Since it is a major result in ...

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### Expected value larger than conditional expected value

Let $X$ and $Y$ be two random variables, with $\mathbb{P}[X] > 5 \cdot \mathbb{P}[Y]$.
Let $h_X(\Delta)$ and $h_Y(\Delta)$ donate the number of hits in a histogram after a time period $\Delta$. In ...

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88
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### First time random sum exceeds value

Suppose $X_n$ $n = 1, 2, \ldots$ are i.i.d random variables with $\mu := \mathbb{E}[X_n]$ > 0. (although they are not necessarily non-negative). Then if $S_n = \sum_{k=1}^n X_k$ and $\tau_a$ = $\...

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### Minimal set of functions to characterize a distribution

In probability theory, there are a number of equivalent ways to characterize a distribution on $\mathbb R^n$. For example, the distribution of a random vector $X\in\mathbb R^n$ may be characterized by:...

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### Probabilistic method Alon and Spencer Azuma's inequality

Theorem 7.5.2 states:
Let $v_1, \dots, v_n$ be vectors with $\|v_i\| \leq 1.$ Let $\epsilon_1, \dots, \epsilon_n \in \{-1, 1\}$ be independent with uniform probability and let $X=\|\epsilon_1 v_1 + \...

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### Control the largest eigenvalue of random matrix

Objective: If there exists $\epsilon > 0$ such that $\sigma <
\sqrt{\frac{n}{(2+\epsilon)\log n}}$
then, with high probability, we have $\lambda_{\text{max}}(D_{[-W]}+W)<\frac{n}{\sigma}$. $W$...

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0
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### Convergence of eigenvalues in the bulk of the GbetaE?

Let $\lambda_{k(n)}$, $k(n) / n \to a \in (0,1)$, be an eigenvalue in the bulk
of the $n \times n$ Gaussian Unitary Ensemble (GUE) normalized so that its spectral measure converges to the semi-circle ...

1
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1
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61
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### An inequality relating $\ell_1$ distance of input and output of a Markov krnel

Let $K$ be a Markov kernel from $\mathcal{X}$ to $\mathcal{Y}$, i.e., $K(\cdot|x)$ is a probability measure on $\mathcal{Y}$ for all $x\in \mathcal{X}$.
Let $\mu$ and $\nu$ be two probability measures ...

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### Lower bound on sum of independent heavy-tailed random variables

I have a sum of $n$ i.i.d random variables $X_i$ such that $E[X_i] = 0$,$\mathrm{E}[|X_i|^{1 + \delta}]$ exists for some $0 < \delta < 1$ but $\mathrm{E}[|X_i|^{1 + \delta+ \epsilon}]$ does not ...

3
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88
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### Inequality: multivariate normal distribition

Let $p(u,x)=\frac{1}{(4\pi u)^{q/2}}e^{-|x|^2/(4u)},u>0,x \in \mathbb{R}^q.$
Prove that for $r\geq 0,c>1$ there exists $C>0$ (depending on $r,c$) such that $$\forall x \in \mathbb{R}^q,u>...

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### Joint distribution of minor of Wigner Hermitian matrices

Let $A$ be an $n\times n$ random matrix with i.i.d entries (say standard Gaussian) $A_{ij}$. I know that there is a CLT type result known for the determinant of $A$. More precisely there is a CLT for $...

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### Bound on inverse covariance from covariance in regularized covariance estimation problem

In this paper by Bickel and Levina, I am confused about result (A15) which claims that since
$$
(A14) \qquad \| \text{Var}(\mathbf{X}) - \widehat{\text{Var}}(\mathbf{X})\|_{\max} = O_P(n^{-1/2} \log^{...

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### Can we further restrict the space of test functions to $C_c^\infty (X)$ in weak convergence?

Let
$X := \mathbb R^n$,
$C_b(X)$ the space of all real-valued bounded continuous,
$C_c(X)$ the space of all real-valued continuous functions with compact supports, and
$C_c^\infty(X)$ the space of ...

3
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### Existence of copula bound pointwise strictly smaller than the Fréchet-Hoeffding upper bound

Consider bivariate copulas $C_1$ and $C_2$ with $\max\{C_1(u,v), C_2(u,v)\}< M_2(u,v)$ for all $u,v \in(0,1)$, where $M_2(u,v) := \min\{u,v\}$ is the Fréchet-Hoeffding upper bound.
Is there a ...

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### Convergence of a non-iid series

Let $\mathbf y \in \mathbb C^N$ be a gaussian random vector with covariance matrix $C_{\mathbf y}$, then $\mathbf x = C_{\mathbf y} ^\frac{-1}{2} \mathbf y$ will be an i.i.d random vector. For a given ...

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### Counting returns in null-recurrent random walk

Consider two independent copies of IID random walk on ${\bf Z}$ starting from $0$, and let $N_1(t)$ (resp. $N_2(t)$) denote the number of times, up to time $t$, that the first (resp. second) walker ...

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2
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86
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### Density of $W_t$ assuming it stayed above a line $L$

Let $W_t$ be a Wiener process with $W_0=0$, and let $L=\{at+by=c\}$ be a line with $c/b<0$ (i.e. the line crosses the $Y$-axis below $0$).
Assume that $W_t$ stayed above $L$ up to time $T$. What is ...

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107
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### Strong law of large numbers when the largest value is deleted

If $X_1, X_2, \ldots X_n, \ldots$ are iid integrable real random variables, and $S_n = X_1 + \cdots + X_n$,
$n \geq 1$, the standard strong law of large numbers expresses that $\frac 1n S_n \to E(X_1)$...

1
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1
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97
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### Stochastic integral with non-anticipating integrand

Let $B$ be a Brownian motion. We want to define $$ \int_{0}^{t} B_{s} dB_{s} : = \lim_{n \to \infty } \sum_{k = 1}^{2^{n}t} B_{\frac{k-1}{2^{n}}}[ B_{\frac{k}{2^{n}}} - B_{\frac{k-1}{2^{n}}}]. $$
To ...

6
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1
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99
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### What's the minimum ratio of positive cells such that the player has a positive probability to reach the boundary of a large random map?

A map with $(2n-1)$ rows and $(2n-1)$ columns of square cells is generated randomly as follows: the value of each cell is chosen from {1, -1} randomly and independently with probability {$p$, $1-p$}. ...

2
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50
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### Finding the limit of a specific sequence of linear processes

Given a strictly stationary random process $(\xi_t)_{t \in \mathbb Z}$. Define $\mu:= E[\xi_t]$, for all $t$. Suppose $(\xi_t)_{t \in \mathbb Z}$ ergodic:
\begin{equation}\label{a}\tag{E}
\frac{1}{n}\...

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446
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### Scheduling "parent talks" at school

Real life motivation. In my younger son's class, there are $18$ students. His teacher provided $18$ time slots for the parents of each child to have a 30-minute conversation of their kid's progress in ...

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40
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### Objective function in denoing diffusion probability model

The objective of the diffusion model can be written as follow (from Lil'Log):
\begin{aligned}
L_\text{VLB}
&= \mathbb{E}_{q(\mathbf{x}_{0:T})} \Big[ \log\frac{q(\mathbf{x}_{1:T}\vert\mathbf{x}_0)}...

4
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216
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### Random walk visiting a cylinder infinitely often

I wonder whether a $d$-dimensional random walk $S_n$, generated by the infinite i.i.d. copies of X given by:
$X=e_1=(1, 0, 0, ..., 0)$ (with probability $p_1$)
$X=e_2=(0, 1, 0, ..., 0)$ (with ...