# 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.

5,175 questions
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### On the distribution of a random point of a poisson process

Let $T = \{t_i\}_{i=1}^\infty$ be the set of points in a Poisson point process on the positive half-axis with parameter $\lambda$, $I \in \mathbb{N}$ be a positive integral random variable with ...
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### Why the Fisher information matrix is equal to the Hessian matrix of the Kullback–Leibler distance at the true parameter?

I'm reading 《Algebraic geometry and statistical learning theory》.My problem is why the Fisher information matrix is equal to the Hessian matrix of the Kullback–Leibler distance at the true parameter?...
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### Exchangeable Bernoulli random variables with bounded summation implies negative correlation?

Let $\big\{X_1, X_2, ..., X_n \big\}$ be $n$ jointly exchangeable Bernoulli random variables, i.e., exchanging the order of these random variables does not change the joint distribution. If we know ...
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### Probability of one species reaching zero before the other in a Markov process on a 2d lattice

$\textbf{Background}$: Say we've got a two-variable system of stochastic chemical reactions, with quantities $\vec{x}(t) = (x_1(t),x_2(t)) \in \mathbb{N}^2$ evolving according to the following system, ...
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### Maximal inequality for the average of i.i.d. random variables

Let $Z_i$ be i.i.d. random variables with $\mathbb{E}[Z_i] = 0$ and $\mathbb{E}|Z_i|^p< \infty$ for $p=1,2,3,\cdots$. I am looking for the following type of estimate if possible, and it is not like ...
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### Is there a transient graph whose spectral dimension two?

Let $G = (V(G), E(G))$ be an infinite connected simple graph. Let $((S_n)_n, (P^x)_{x \in V(G)})$ be the simple random walk on $G$. Let $p_n (x,y) = P^x (S_n = y)$. A spectral dimension of $G$ is ...
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### Concentration of closed random walks

Consider a random walk $S_n=\sum_{i=1}^n X_i$ where $P(X_i=+1)=P(X_i=-1)=1/2$ with $n$ large. By Chernoff's bound we know that, for example, $\sum_{i=1}^{n/2} X_i=O(\sqrt{n})$ with high probability. ...
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### Using common samples to numerically estimate pairwise equality of three random variables

Let $X,Y,Z$ be three discrete random variables which I can numerically sample. I need to numerically estimate the probability that $X=Y$ and the probability that $X=Z$. I would like to know whether ...
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### Sum of independent random walks

Given two independent random walks $S$ and $S'$ with different distributions for the random variables $X_1$ and $X_1'$, I am interested in studying the conditions that make their sum either a ...
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### Generalization of inverse transform sampling

If X is a random variable over an arbitrary alphabet, is there a (deterministic) function f() such that X = f(U), where U is a uniform random variable over the unit-interval?
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### Expected value of eigenvalue of matrix

Let $A = (X_{ij})_{ij}$ a square matrix of size $n$ where the $X_{ij}$ are (discrete) real random non-negative entries. Denote by $\lambda_1(A) \geq \dots \geq \lambda_n(A)$ the (random) ordered ...
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### Randomly picking $k$ members of $\{1,\ldots,n\}$

Every day, I randomly pick a sample consisting of $k$ members of $\{1,\ldots,n\}$ where $k\leq n$. I stop as soon as every number of $\{1,\ldots,n\}$ has been picked at least once. Let $S$ be the ...
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Let $(\Omega,\mathcal A,\operatorname P)$ and $(E,\mathcal E,\pi)$ be probability spaces $(X_n)_{n\in\mathbb N}$ be a $(E,\mathcal E)$-valued stochastic process on $(\Omega,\mathcal A,\operatorname P)... 1answer 46 views ### Lyapunov-type function in a non locally-compact space and boundedness of the average Set-up and question. Let$\mathcal{X}$be a complete separable metric space which is not locally-compact. Let$V: \mathcal{X} \to [0; +\infty]$be a function and$(X_t)_{t\geq 0}$a Markov process in$...
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Is it true, and if so, is there an easy way to see that the quadratic variation of standard Brownian motion on n-sphere is $\leq$ t? Note: I am a novice in stochastic analysis.
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### Compactness if Emery (semi-martingale) topology

The set $\mathscr{S}$, of semi-martingales is a topological vector space under the Emery topology on the space of semi-martingales. There has been some recent research on closures in this topology (...
### Random walk on $\mathbb{R}$ with “sticky” origin
Let $P_i$, $N_i$, and $Z_i$, $i\in\mathbb{N}$ be r.v.'s with the $P_i$, $N_i$, and $Z_i$ being identically distributed with known pdf's $f_P$, $f_N$, and $f_Z$, respectively; and with no dependence ...