# 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.

Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

7,901
questions

0
votes

0
answers

34
views

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

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\...

0
votes

0
answers

34
views

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

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 $S$, ...

4
votes

0
answers

151
views

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

0
answers

72
views

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

0
answers

82
views

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 $a > 0$.
...

5
votes

1
answer

200
views

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 $...

1
vote

0
answers

52
views

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 ...

0
votes

0
answers

23
views

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+\...

10
votes

0
answers

513
views

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, ...

0
votes

1
answer

73
views

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_{...

5
votes

0
answers

50
views

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 ...

1
vote

0
answers

23
views

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 +...

0
votes

1
answer

51
views

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

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

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(...

1
vote

1
answer

291
views

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 ...

1
vote

0
answers

115
views

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 \...

0
votes

0
answers

60
views

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 $\sigma$-...

1
vote

1
answer

301
views

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

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 ...

1
vote

0
answers

105
views

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 $\...

0
votes

0
answers

62
views

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,...

1
vote

1
answer

110
views

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 ...

0
votes

0
answers

155
views

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

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_\...

2
votes

1
answer

65
views

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 ...

0
votes

1
answer

64
views

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)}...

1
vote

0
answers

49
views

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 ...

1
vote

0
answers

76
views

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

0
votes

0
answers

59
views

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,\...

3
votes

1
answer

509
views

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

0
answers

81
views

$\DeclareMathOperator\Ent{Ent}$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

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=\...

0
votes

0
answers

43
views

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 ...

0
votes

1
answer

121
views

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

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

"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

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

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

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 ...

1
vote

1
answer

71
views

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 ...

1
vote

1
answer

66
views

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

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

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

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

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

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 ...

0
votes

0
answers

31
views

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 ...