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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Can I explore the infinite cluster of Bernoulli percolation in $\mathbb{Z}^2$?

In Bernoulli percolation on the square lattice $\mathbb{Z}^2$ every edge is kept with a probability $q$ and erased with probability $1-q$. It is classical that whenever $q > \frac{1}{2}$ there is ...
5 votes
1 answer
154 views

Maximal inequality of iid random variables $\{X_{ij}\}_{1\leqslant i,j \leqslant n}$

Suppose that $\{X_{ij}\}_{1\leqslant i,j\leqslant n}$ are iid random variables with $\mathbb{E}(X_{11})=0$ and $\mathrm{Var}(X_{11})=1$, does the following convergence hold: $$ \max_{1\leqslant j\...
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Expectation of a norm in Monte-Carlo method

I am currently studying the Monte Carlo methods for solving PDEs with random coefficients. My problem here is basically just doing with some algebraic properties of the expected value function which I ...
1 vote
1 answer
66 views

Gaussian width of intersection of cube and ball in high-dimensional euclidean space

Let $d$ be a large positive integer and fix $r \ge 0$. Set $S := B_2^n \cap [-r,r]^d$, where $B_2^d$ is the euclidean unit-ball in $\mathbb R^d$. Finally, let $\omega(S)$ be the Gaussian width of $S$, ...
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Asymptotic behavior of a dynamical system of density functions

On September 24, 2022, I asked the question below on Mathematics Stack Exchange, linked here: Link to question on Mathematics Stack Exchange. I received two up-votes, but no comments or answer. I ...
3 votes
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72 views

Bounds on the expectation of a product of zonal spherical harmonics

Let us consider a $d-1$ dimensional sphere $S^{d-1}$, and for a point $a \in S^{d-1}$ let $Z_{a,k} : S^{d-1} \to \mathbb{R}$ be the zonal spherical harmonic of degree $k$ in the direction $a$, with ...
3 votes
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82 views

Conditional distribution of steps of random walk given the sum

Set-up. Consider a random walk $S_n=\sum_{i=1}^n X_i$, where $\{ X_i, 1\leq i < \infty \} $ is a sequence of i.i.d. random variables with distribution $\mu$, $\mathbb{E}X_1 = 0$. Let $a > 0$. ...
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200 views

Abstract characterization of white noise

Let $\xi$ be a random variable valued in the space of Schwartz distributions $\mathcal{S}'(\mathbb{R}^d)$. For any open set $R\subset\mathbb{R}^d$ let $\Sigma(R)$ be the $\sigma$-algebra generated by $...
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Is this formula for the optimal test statistic well-known?

Consider a random vector $X$ in $\mathbb R^n$. Under the null hypothesis, $X$ has probability density distribution $f$. However, the experimenter has a strong suspicion that $X$ actually satisfies ...
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Inhomogeneous Markov chains and the product-integral as a solution to the Kolmogorov forward equation

We have a inhomogeneous continous $K$-State Markov chain $X(t)$ with transition intensity matrix $Q(t)$. Therefore its entries are: $$q_{ij}(t)= \lim_{\delta \to 0} \frac{1}{\delta} \mathbb{P}(X(t+\...
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A simple stochastic game

An individual, henceforth called the runner starts at the center of an open two dimensional square $\Omega$ of side length $r \geq 2$. At each turn, a vector $x \in S^1$ is chosen uniformly at random, ...
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Change of measure formula for the Föllmer process

While reading a preprint Eldan, Lehec, and Shenfeld - Stability of the logarithmic Sobolev inequality via the Föllmer Process I came across the following SDE in Section 3: $$d X_t=d B_t+\nabla \log P_{...
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Criteria for tightness of Gaussian measures on Banach spaces

In Bogachev's book "Gaussian Measures" (Example 3.8.13) sufficient conditions for the (uniform) tightness of a sequence of centered Borel Gaussian probability measures on a separable Hilbert ...
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Subordinated non-deterministic Gaussian process is non-deterministic

Let $X = \{ X(k), k \in \mathbb{Z} \}$ be a strictly stationary, Gaussian time series whose spectral density $f_X$ exists. Furthermore, let $X$ be non-deterministic, i.e. $$ \mathbb{E}\big[ \vert X(n +...
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Rates of convergence of empirical measures in Wasserstein distance

Let $X_1, X_2, \ldots$ be iid random variables in $\mathbb {R}^d$ with common distribution $\mu$, and $\mu_N = \frac 1N \sum_{k=1}^N \delta_{X_k}$, $N \geq 1$, the associated empirical measures. If $\...
11 votes
1 answer
934 views

Can deleting a random entry from an iid sequence destroy the iid property?

Let $X=(X_1,\ldots,X_n)$ be an iid sequence of random variables, and let $\nu$ be a uniformly random integer in the range $1,\ldots,n$. Then $\xi_\nu$ is a random entry of $X$. Is it always true that ...
7 votes
1 answer
136 views

Onsager-Machlup functional when drift is time-dependent

Let $X(t)$ be a diffusion process on $\mathbb{R}^d$ generated by \begin{align} \mathcal{D} = \nabla^2 + \sum_{i=1}^d b_i(x) \frac{\partial}{\partial x_i}, \end{align} where $b_i(x) \in \mathcal{C}_b^2(...
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1 answer
291 views

How to calculate this limit (if exist)?

I have just asked the calculation of the following summation see here $$S(a,b,m,n_1,n_2)=\sum_{k=0}^m a^k b^{m-k} {n_1\choose k} {n_2\choose m-k}, $$ which is motivated by the calculation of the ...
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115 views

An estimate for the first eigenfunction of the fractional Laplacian and a kind of "maximum principle"

Let $\Omega$ be a bounded Lipschitz domain. Let $u$ be the first eigenfunction of the fractional Laplacian $$ (-\Delta)^s u = \lambda u \ \text{ in } \Omega, \quad u = 0 \ \text{ in } \mathbb R^n \...
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Regularity of map from $L^p$ to $\mathcal{W}_p$

Let $(X,d)$ be a polish metric space, fix a probability measure $\mathbb{P}$ on $(X,d)$ belonging to the Wasserstein $\mathcal{W}_p(X,d)$ for some fixed $p\in [1,\infty)$. Denote the Borel $\sigma$-...
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1 vote
1 answer
301 views

Maximum of a sequence is $o(\sqrt{n})$

Quick context: This is a transposition of exercise 6.3 of Remco Van der Hofstad (2016) and it is relevant to some problem i encoutered in my research. For each $n \in \mathbb{N}$, define a series of ...
1 vote
1 answer
76 views

Popular algorithms (stopping rules) with output - a prefix of a permutation

What are some popular settings, when we look at the elements of a randomly generated permutation one by one, and we use certain stopping rule which, as a result gives us a prefix of the observed ...
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105 views

Hardy's inequality proof using Doob's inequalities

Consider a probability space $([0,1],\mathcal{B}([0,1],\lambda),p>1$ and $f \in L^p(]0,\infty[).$ We want to prove Hardy's inequality using martingale theory and Doob's maximal inequalities. Let $\...
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Ito-Tanaka formula for SPDEs

I found a Tanaka's formula proved by Briand, et al in (Stochastic Processes and their Application, 108 (2003) 109-129), that is, Let $\left\{K_t\right\}_{t \in[0, T]}$ and $\left\{H_t\right\}_{t \in[0,...
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1 vote
1 answer
110 views

Proof of lower bound on variance

I'm reading through the paper Poincaré type and spectral gap inequalities with fractional Laplacians on Hamming cube. However, I'm having a difficult time understanding the following proof: Lemma 2.1 ...
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155 views

Does $E[1/f]\overset{d}\to 1/E[f]$ for power-law spectra with fixed second moment?

Edit in earlier version of this question it was suggested that observed behavior is a consequence of concentration property, however, Iosif Pinelis showed that desired concentration property will ...
4 votes
0 answers
82 views

(Asymptotic) Cycle structure in a random permutation given total number of cycles?

A random $N$-permutation is the one drawn uniformly from all possible permutations on $N$ points. We know that the expected number of cycles of length $\ell$ in a random $N$-permutation, $\mathbb{E}C_\...
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1 answer
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What condition on random matrix can preserve sub-Gaussian property?

Suppose $x\in SG(\sigma^2)$ is a sub-Gaussian random vector, i.e. $\left<u,x\right>\quad \forall u\in \mathbb{S}^{n-1}$ is a sub-Gaussian random variable. My question is : what condition on the ...
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64 views

WLLN for bootstrap means of stationary ergodic processes?

Setup:$\quad$ Suppose that $(X_n)$ is a stationary ergodic process with $E|X_1|<\infty$. Given $X^{(n)}=(X_1, \dots, X_n)$, select a standard Efron bootstrap subsample $(X_{n,1}^*, \dots, X_{n,m(n)}...
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1 vote
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Concavity of expected size of a maximum matching (in a bipartite graph) w.r.t. edge probability

Given a n*n bipartite graph where each edge (between any two nodes on the opposite side) is formed i.i.d. with probability $p$, can we show a concavity result on the expected size of a maximum ...
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1 vote
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76 views

Limit of alternating sum of factorial moments which diverge

Consider the non-negative, integer valued random variable $X$, and its $i^{\text{th}}$ factorial moment $E_{i}[X]$. Then we have that $$ P(X=0) = \sum _{i=0}^{\infty} \frac{(-1)^i E_{r}[X]}{ i!} $$ ...
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59 views

A moment-based stochastic order

Given that a Borel probability measure $\mu$ on [0,1] is characterized by its moments, it seems natural to consider the following stochastic order: say that $\mu\le\mu'$ if $$\forall k\in\mathbb N,\...
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3 votes
1 answer
509 views

Does $E[1/f]\overset{d}\to 1/E[f]$ for $\operatorname{Tr}H=1,\operatorname{Tr}H^2=0.5$?

Suppose $x$ is a Gaussian random variable in $d$ dimensions with $H=E[xx^T],\ \operatorname{Tr}(H)=1,\operatorname{Tr}(H^2)=0.5$. Take $m$ I.I.D. samples of $x$ and stack them as rows of $X$. Is it ...
4 votes
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81 views

Weighted logarithmic Sobolev inequality

$\DeclareMathOperator\Ent{Ent}$The usual logarithmic Sobolev inequality says that $$ \Ent_\mu(f^2)\leq C\int |\nabla f|^2 d\mu $$ where the entropy $$ \Ent_\mu(f^2)=\int f^2 \log\left( \frac{f^2}{\int ...
1 vote
1 answer
258 views

How fast does this Gaussian random walk move away from the origin?

Suppose $z_i$ are IID zero-centered $d$-dimensional Gaussian random variables with unit-trace covariance $\Sigma$ and $g(z_i)$ is the sum of its components. Consider the following random walk: $$x_s=\...
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Continuity of exit times on path spaces

Let $(E,d)$ be a locally compact separable metric space. Let $\mathcal{D}=\mathcal{D}([0,\infty),E)$ denote the space of right continuous functions on $[0,\infty)$ having left limits and taking ...
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Comparison of Rademacher and Gaussian expected values under linear transformations

As per suggestion, I have decided to post the following as a new question, but it is a follow-up to this one: Comparison of Rademacher and Gaussian moments under linear transformations Let $X$ be an $...
1 vote
1 answer
73 views

Range of $a$ such that $w \leftarrow w-a x \langle w, x \rangle$ converges almost surely?

Suppose $x_i$ come from 2-d standard Normal centered at 0. What is the range of $a$ for which the following iteration converges almost surely? $$w_{i+1} = w_i-a x_i \langle w_i, x_i \rangle$$ For 1-d, ...
6 votes
2 answers
2k views

Expected maximum number of "prank cigarettes" in an average pack

"Real-life" motivation. The German satirical magazine Der Postillon suggested a few measures for deterring smokers from their bad habit. I especially liked the idea of inserting one "...
2 votes
1 answer
113 views

$\Psi$ in finite Wiener–Itô Chaos implies existence of continuous representative on neighborhood of Cameron–Martin space?

Let $(\Theta, H, \mu)$ be an abstract Wiener space, i.e. let $(\Theta, \lVert \cdot \rVert_{\Theta})$ be a separable Banach space, let $(H, \langle \cdot, \cdot \rangle_{H})$ be a separable Hilbert ...
3 votes
1 answer
124 views

Random covering of a set

Let $A$ be a set of $n$ elements. Let $S_1,\dots,S_n$ be independent $k$-element random subsets of $A$. What is the probability that $S_1,\dots, S_n$ evenly cover $A$, i.e. each element of $A$ belongs ...
1 vote
2 answers
80 views

Choosing $k$ different assignments of binary variables in order to capture the largest volume of the joint probability distribution

Assume you have $n$ independent binary variables $\{x_1,\dots,x_n\}$ and for each variable $x_i$ you know that its value is equal to $1$ with a probability $p_i$. I would like to enumerate the joint ...
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1 vote
1 answer
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What happens in the difference rate between these two versions of ballot theorem?

Here are two different versions of Gaussian ballot theorems in the literature, each on different while similar events but the rate is quite different: P39, Probability Result 1: For any independent ...
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1 vote
1 answer
66 views

Comparison of Rademacher and Gaussian moments under linear transformations

Let $X$ be an $n$ dimensional standard Gaussian and let $U$ be an $n \times n$ orthogonal matrix. Then, the random vector $Z = U^\top X$ is also distributed as a standard Gaussian in $R^n$ and we have ...
3 votes
1 answer
130 views

Probabilistic Taylor theorem for concave functions

This paper proves a probabilistic version of Taylor's theorem \begin{equation*} \mathbb{E}g(X) = \sum_{k=0}^{n-1} \frac{g^{(k)}(0)}{k!} \mathbb{E}X^k + \frac{\mathbb{E}X^n}{n!} \mathbb{E} g^{(n)}(X_{(...
5 votes
0 answers
116 views

Particles sent into the same direction with uniformly distributed speed

Fix a positive integer $n$. Every second, a particle is sent along a straight line from a fixed position in a fixed direction, at a random integer speed chosen uniformly in $\{1,\ldots, n\}$ meters ...
1 vote
0 answers
63 views

Rough path expected signature vs cumulant-generating function / characteristic function

What is the point of using rough path expected signature to characterize the law of а stochastic process when the cumulant generating function is known ($\log\mathbb{E}[e^{i\theta X(t)}]$)? Since an ...
3 votes
0 answers
184 views

High-dimensional uniform distribution

Let $\mathcal X$ be either a subset of $\mathbb R$ equipped with Lebesgue measure or a countable set with counting measure. The Gibbs' principle in statistical physics asserts that if $(X_1 , \dots, ...
29 votes
4 answers
1k views

Expected length of longest stick in a stick snapping process

Starting with a single stick of unit length, a point $p \in (0, 1)$ is picked uniformly at random along the stick and the stick is snapped, producing two sticks of length $p$ and $1-p$. At each next ...
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0 votes
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31 views

Distribution of class sizes above a certain threshold in a stick-breaking process

Given a stick-breaking process, that is to say a sequence of random variables $(Y_n)_n$ such that $$Y_n = X_n \prod_{i=1}^{n-1}(1 - X_i)$$ where the $X_n$ are independent random variables distributed ...