# 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 ...
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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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### 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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### 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 ...
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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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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, ... • 2,102 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 ... • 150 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 ... • 953 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 ... • 13 1 vote 1 answer 71 views ### 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 ... • 547 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, ...
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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 ...
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 ...