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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63 views

Eigenvalue distribution of random matrices

Let $M_{m \times n}$ be a matrix with real entries $m_{ij}$ which are Gaussian independently distributed. If we further assume that $m = n$, $m_{ii} \sim N(0, 2)$, $m_{ij} \sim N(0, 1), i\neq j$, and ...
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1answer
64 views

“Relative compactness of a family of probability measures” and relative compactness & sequential compactness of sets

I'm studying Billingsley's convergence of probability measures, and wondering why the definition of "Relative compactness of a family of probability measures" reasonable. In the discussion ...
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2answers
308 views

Question about a new pseudo-random number generator

While investigating non-periodic RNG's (random number generators) for irrational numbers, I came up with a version that actually produces pseudo-random words consisting of $N$ bits, where $N$ is ...
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48 views

Are stable matchings (noise-)stable?

Suppose a group of computer scientists have entrusted their dating lives to a computer. Specifically, there are $n$ men and $n$ women, all of whom are cis-het. Being educated people, they of course ...
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50 views

Motivation for proof of local lemma/construction version

I am interested in finding intuition to the bounds and proof of the asymmetric local lemma. I think the $k$-SAT is fairly intuitive, but I would like to understand the general version. One good ...
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62 views

approximate the square of 2-norm distance between binary distributions with high probability

Suppsose we take $m$ samples from a Bernoulli distribution with probability $p$, and $m$ samples from another probability distribution with probability $q$. We want to calculate a statistic $x$ from ...
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52 views

The Itō isometry for Riemannian manifolds

If $\alpha$ is a real smooth $1$-form, and if $\mathcal C$ is the space of continuous functions $c : [0,1] \to \mathbb R^n$, endowed with the Wiener measure $w$, and if $I_\alpha : \mathcal C \to \...
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43 views

Independent increments for the Brownian motion on a Riemannian manifold

In am not a probabilist, but I must do some stochastic-flavoured work on a connected Riemannian manifold $M$. A nice thing about the Brownian motion on $\mathbb R^n$ is that we may talk about its ...
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241 views

Measuring the randomness of texts

The question concerns statistic properties of random words in a finite alphabet $A$. By $A^{<\omega}$ we denote the set of all words in the alphabet $A$, i.e. finite sequences of elements of $A$. ...
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177 views

Is there a practically useful or concrete representation theory/Fourier analysis on finite groupoids?

Fourier analysis on finite groups is well known to be useful for probability theory and combinatorics — consider for example the Fourier analysis on $(\mathbb Z/2\mathbb Z)^n$ which can be used to get ...
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2answers
101 views

Induced probability measure on a finite orbit under a group action

Suppose we have a discrete group $G$ acting on a compact set $X \subseteq \mathbb{R}^d$ via measure-preserving homeomorphisms, and suppose we have a point $x$ whose orbit $Gx$ is finite (say $|Gx| = n$...
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1answer
166 views

Is there a standard way of defining the integral of an extended real function with respect to a finitely additive probability measure?

Let $X$ be a set, and let $\mu$ be a finitely additive probability measure defined on $2^X$. Let $\Phi$ be the set of functions from $X$ to $\mathbb R \cup \{-\infty, \infty\}$. Is there a standard ...
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81 views

Can one define a bounded noise process by conditioning standard Gaussian white noise on the assumption of boundedness?

Background of the question. One of the problems that arises with Wiener-driven (and more general Lévy-driven) models of noisy systems is that, due to the extremely rapid decay of tails of infinitely ...
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82 views

Is there a name for a random variable that is the absolute value of the difference between two iid discrete uniform variables?

I'm working on a project and I needed to calculate the distribution of the difference between two iid discrete uniform variables (sorry for the long title). That is, let $I, J$ be two iid discrete ...
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1answer
74 views

Inequality for integrals of probability densities

Given three univariate probability densities, $f(x)$, $g(x)$, and $h(x)$, I would like to show that $\int_{supp(f)\cap supp(g)}\frac{fg}{f+g}dx+\int_{supp(g)\cap supp(h)}\frac{gh}{g+h}dx-\int_{supp(f)\...
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1answer
343 views

Symmetric distribution optimization problem of distances between points in the interval $[0,1]$

Let $\mathcal{D}$ be a probability distribution with support $[0,1]$. Let $x, y, z$ the outcomes of three i.i.d. random variables $X, Y, Z$ with distribution $\mathcal{D}$, sorted in increasing order, ...
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34 views

Semimartingale decomposition and filtrations

In short: I am trying to understand how the decomposition of a semimartingale into its local martingale and finite variation components depends on the filtration we are using. So, taking a toy example,...
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25 views

Linear independence of Wishart matrices

Let $W\sim W_n(I,d)$ be a real Wishart matrix of an identity covariance matrix and $d$ degrees of freedom, i.e., $W=XX^T$ for $X$ being an $n\times d$ matrix whose entries are i.i.d sampled from a ...
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1answer
62 views

Fourth moment of a random-variable with block-tridiagonal structure

Let x be a random variable in $\mathbb{R}^d$, $J$ a block tridiagonal $d\times d$ matrix, and probability of $x$ is defined as follows $$p(x)\propto \exp(-x'Jx)$$ For a fixed $d\times d$ matrix $v$ ...
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47 views

Discretization formula for system of two differential equations. “Solution to one of these is the initial condition of the other”. In which sense? [closed]

Consider the following stochastic differential equation \begin{equation} dy=\left(A-\left(A+B\right)y\right)dt+C\sqrt{y\left(1-y\right)}dW\tag{1} \end{equation} where $A$, $B$ and $C$ are parameters ...
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47 views

Expectation of angle between two vectors in the image of a gaussian random matrix

Let $m$ and $n$ be large positive integers (going to infinity), and let $W$ be a random matrix of size $n \times m$ with iid entries from $N(0,1/m)$. Let $x,y \in \mathbb R^m$ be deterministic vectors,...
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2answers
72 views

Non-uniqueness of loop-erasure for continuous-time curves

Question. Is there a continuous curve in the plane that has a non-unique loop-erasure? Here is the definition of a loop-erasure. A continuous curve $Y:[c,d]\to\mathbb R^2$ is a loop-erasure of a curve ...
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2answers
159 views

Exponential inequality for the sum of martingale differences $X_1, \dots, X_n$ when $\sum_{i=1}^{n} \operatorname{Var}(X_i) \leq B^2$

Let $X_1, X_2, \dots, X_n$ be a martingale difference sequence such that $$ X_i \leq y \quad \text{and} \quad \sum_{i=1}^{n} \operatorname{Var}(X_i) \leq B^2. $$ Question 1: Does the following hold? $$...
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26 views

(random fields / gaussian process): On rewritting a certain expectation as a kernel function

Let $v = (v_1,\ldots,v_n)$ and $(w_{1,1},\ldots,w_{1,n},\ldots, w_{n,m})$ be random vectors with iid coordinates, and also $v$ is independent of $w$, with $w_{i,j} \sim N(0,1/m)$ and $v_j \sim N(0,1/n)...
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1answer
162 views

Probability distribution optimization problem of distances between points in the interval $[0,1]$

Let $\mathcal{D}$ be a probability distribution with support $[0,1]$. Let $X, Y, Z$ three i.i.d. random variables with distribution $\mathcal{D}$, and $T$ a random variable uniformly distributed in $[...
1
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2answers
56 views

Expectation of the determinant of the inverse of non-central Wishart matrix

Let $A$ be $(n,n)$ central Wishart matrix with $k$ degrees of freedom. my question is there is a way to estimate the expectation of: \begin{align} E[det(I+(I+A)^{-1})] \end{align}
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1answer
56 views

Compute limit of $\mathbb P(Y \le X_n)$ using limiting information on the sequence of random variables $X_n$

Let $Y$ be a symmetric random variable, $(X_n)_n$ be a sequence of nonnegative random variables, and set $p_n = \mathbb P(Y \le X_n)$. It is known from Slutsky's theorem that, if $c$ is a constant ...
4
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1answer
80 views

When does a gaussian quadratic form converge (in probability) to a constant?

Let $(h_{ij})_{i,j \in \mathbb N}$ be a sequence of real numbers (deterministic) and let $x_1,\ldots,x_n,\ldots$ be a sequence of iid $N(0,1)$ randm variables. For each positive integer $n$, consider ...
3
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1answer
93 views

Kac-Rice formula and Borell-TIS inequalities for gradient-flow of centered gaussian random field

Let $x\mapsto g(x)$ be a centered gaussian random field on $\mathbb R^m$. Let $x_0 \in \mathbb R^n$, and (assuming regularity conditions) consider the gradient-flow $$ \dot{x}(t) = -\nabla g(x(t)), \;...
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0answers
94 views

Intersection of a Poisson bridge and a Brownian bridge

Take a Poisson process $N_t$, a Brownian motion $W_t$ and constants $T > 0$ and $a > 0$. Suppose $N$ and $W$ are independent. I'm interested in the probability that $W$ does not cross over $a + ...
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1answer
59 views

Central limit theorem for chi-squared random field on $\mathbb R^p$

Let $X:x \mapsto X(x)$ be a centered stationary Gaussian process on the $\Omega:=\mathbb R^p$, such that $X(x) \overset{d}{=}X(x')$ for all $x,x' \in \Omega$. Set $\sigma^2 := \mbox{Var}(X(0)) = \...
1
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1answer
65 views

Concentration inequality for the supremum of $L_2$ norm of a vector-valued Gaussian process with iid components

Let $\Omega$ be a compact subset of $\mathbb R^p$ and let $f_1,\ldots,f_k$ be zero mean identically distrubuted Gaussian processes on $\Omega$ such that $f_1(x),\ldots,f_k(x)$ are independent $x \in \...
3
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1answer
71 views

Characterizing 'very homogeneous' finitely valued stochastic processes

Fix a positive integer $n$. Let $X = \{X_i\}_{i \in \mathbb{N}}$ be a discrete time stochastic process such that each $X_i$ is a $\{0,\dots,n-1\}$-valued random variable. Suppose that the joint ...
1
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1answer
48 views

Characterization of random variables whose tensor powers have subexponential “small-ball” probabilities

Is there a succinct characterization of all random variables $\zeta$ on $\mathbb R$ with the following properties 1. Symmetry: $\zeta \overset{d}{=} - \zeta$. 2. Small-ball probability: there exists ...
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3answers
453 views

Taking points uniformly inside a general finite geometric domain

It is well known that if we want to take $n$ uniformly and randomly points inside a circle of radius $r$ and centered at the origin the following apparently correct approach for generating $x$ and $...
2
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2answers
525 views

Recursive random number generator based on irrational numbers

Here $\{\cdot\}$ and $\lfloor \cdot\rfloor$ denote the fractional part and floor functions respectively. For a negative, non-integer number $x$, we use the following definition: $\{x\}=1-\{-x\}$. If $...
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123 views

Why are financial markets modeled by càdlàg processes?

When opening a book or reading an article on mathematical finance, financial markets (e.g. stock prices) are always modeled by càdlàg semimartingales. I was wondering why it is that these processes ...
2
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1answer
139 views

Prove / disprove: If $1 \le n < N$ and $A$ is an $N \times n$ matrix with iid from $\mathcal N(0,1)$, then $s_\min(A) \ge c\sqrt{N}$ w.p $1-2e^{-N}$

Let $1 \le n < N$ be integers and $A$ be a random $N\times n$ matrix with iid entries from $\mathcal N(0,1)$. This paper (Rudelson and Vershynin) claims in the paragraph just before formula (3.4) ...
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1answer
55 views

Diagonal terms in the Kochen Stone inequality

In a paper in Lecture Notes in Mathematics vol. 1874, Yan states the Kochen-Stone theorem in the following form, where $A_n$ is a sequence of events such that $\sum_{n=1}^\infty P(A_n) = \infty$: $$ ...
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1answer
167 views

When is a function on symmetric positive definite matrices an expectation of Gaussian?

Is there some characterization of real-valued functions of the form $\Phi(C)=\mathbb{E}F(X)$, where $X$ has the Gaussian $N(0,C)$ distribution, on the space of symmetric positive semidefinite $n\times ...
4
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1answer
82 views

Covering of discrete probability measures

Let $\mathcal{P}_{n:+}(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum_{i=1}^n k_i \delta_{x_i}$ where $k_i>0$. Then any measure in $\mathcal{P}_{n:+}(\...
4
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1answer
63 views

Continuous selection parameterizing discrete measures

Let $\mathcal{P}_n(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum_{i=1}^n k_i \delta_{x_i}$. Then any measure in $\mathcal{P}_n(\mathbb{R})$ is in the image of ...
0
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1answer
68 views

Lower-bound on smallest singular-value of rectangular random matrix

Let $X$ be a random $N \times n$ matrix with iid entries from $\mathcal N(0, 1)$ and with $n/N =: \lambda(N,n) \le \lambda_0$, for some $\lambda_0 \in (0, 1)$. That is, $X$ is genuinely rectangular (...
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1answer
52 views

A scaled random walk on the number line

An agent $A$ is performing a random walk on the number line. Let $X_t$ be his position at time $t$. $X_{t+1}$ is calculated according to the following rules:- $ X_{t+1} =$ \begin{cases} ...
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0answers
79 views

Expectation of the trace of $(AB)^2$ as function trace of $AB$ [closed]

Let $A$ and $B$ be a random square matrix. Is it possible to express $E[\operatorname{trace}((AB))]^2$ by means of $E[\operatorname{trace}(AB)]$ ? or at least something close?
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54 views

Ergodicity properties of a linear operator on the Wiener Space

Suppose I have a centered Gaussian measure $\mu$ on a separable Banach space $B$. Let $H \subset B$ be its Cameron-Martin space and assume that it is dense in $B$. Suppose that $U:H \to H$ is a linear ...
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0answers
151 views

Does additive Gaussian noise preserves the Shannon entropy ordering?

Suppose that $Z$ is a Gaussian random variable independent of $X$ and $Y$. Moreover suppose that $h(X) \geq h(Y)$, where $h(\cdot)$ is the differential Shannon entropy. Does relation $h(X+Z) \geq h(Y+...
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1answer
90 views

Expectation of random matrix

Assume $Q$ is a positive definite random matrix such that $0 < \lambda_{\min}(Q)....\leq \lambda_{\max}(Q) \leq 1$ holds. I want to show that \begin{align} E\left[\frac{\lambda_{\min}(Q)}{\lambda_{\...
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0answers
38 views

Anti-concentration of measure: Slud's inequality for finite populations

Suppose that I have Bernoulli trials with unknown bias $p$. I need $\Omega(\frac{\log 1/\delta}{\epsilon^2})$ samples for the average of the samples to estimate $p$ within $\epsilon$ error with ...
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0answers
38 views

A local base for space of probability measures with Prohorov metric

Let $S$ be a Polish space. Let $P(S)$ denote the space of probability measures on $(S,\mathcal{B})$, where $\mathcal B$ is the Borel-$\sigma$-algebra over $S$. Equip $P(S)$ with the Prohorov metric. I ...