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.

Filter by
Sorted by
Tagged with
4 votes
3 answers
198 views

Expected distance of nearest matching pair in the game of pairs

Recently I was playing several rounds of the game of pairs with my children. I was surprised that almost every time, one matching pair was adjacent (either next to each other in a row, or vertically). ...
Dominic van der Zypen's user avatar
0 votes
1 answer
152 views

Shannon problem

Since a few days, I try in my research to model / formalize a source of Shannon a little weird, and I can't do it at all. First of all, I explain to you its operating principle and then I describe it ...
lulu2612's user avatar
0 votes
0 answers
138 views

Statistical models of functions

I did a quick literature search and found nothing on "statistical models of functions". Let me explain what I am looking for. Given the category of Sets and Function, we have arbitrary functions ...
Ben Sprott's user avatar
  • 1,281
1 vote
1 answer
326 views

Expected values of two non-negative, integer-valued random variables related to an urn problem

Consider an urn containing $c$ distinguishable balls, $\alpha$ of which are red, $\beta$ of which are blue, and $\gamma$ of which are green, and $\alpha+\beta+\gamma=c$. We assume $\alpha,\beta,\gamma&...
Andrea Prunotto's user avatar
4 votes
1 answer
369 views

Is a Gaussian measure on a Hilbert space determined by the coarser topology induced by the covariance operator?

I have a basic question about Gaussian measures on a Hilbert space: Let $\mu$ be a non-degenerate Gaussian measure on a Hilbert space $(H_0,\left\langle \cdot,\cdot \right\rangle_0)$. Then the ...
r_faszanatas's user avatar
2 votes
0 answers
391 views

How can we treat the generator of a discrete-time Markov chain as the generator of a Markov-jump process?

In the popular paper Weak Convergence and Optimal Scaling of Random Walk Metropolis Algorithms by Roberts, Gelman and Gilks, the authors state (see below) that "in the Skorokhod topology, it does not ...
0xbadf00d's user avatar
  • 161
3 votes
1 answer
182 views

Is there a coupling that induces a given coupling via a transition kernel?

Let $X,Y$ be two measurable spaces, $\mu,\nu$ two probability measures on $X$, and $\kappa$ a transition kernel from $X$ to $Y$. Define $\tilde\mu(dy)=\int_X\kappa(dy|x)\mu(dx)$ and $\tilde\nu(dy)=\...
S.Surace's user avatar
  • 1,675
12 votes
1 answer
555 views

Mode of a sum of Bernoulli random variables

Let $S_n=\tau_1+\cdots+\tau_n$ be a sum of independent Bernoulli random variables such that $\mathbb{P}(\tau_i=1)=p_i$. Is it true that the mode of $S_n$ is either its mean rounded up or rounded down?
TOM's user avatar
  • 2,218
3 votes
0 answers
152 views

Probability distribution from equidistribution - I

Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$. Denote $N_r(a,b)$ to be minimum $\ell_r$ norm of vector $(x,y)$ as $(x,y)$ ranges over all non-zero integral solutions to $(x,y)\equiv t(a,...
Turbo's user avatar
  • 13.7k
3 votes
1 answer
115 views

Probability density from standard domain - I

Pick $x+iy$ at random with respect to hyperbolic measure from $\{z:|z|\geq1,|\mathcal R(z)|\leq\frac12\}$. What does the probability distribution function of $\frac1{\sqrt y}$ look like?
Turbo's user avatar
  • 13.7k
3 votes
2 answers
180 views

Is the covariance of squares always bounded from below by two times the covariance?

I came across the following inequality in one of my calculations ($X,Y$ are centered random variables): $$\operatorname{E}(X^2Y^2)-\operatorname{E}(X^2)\operatorname{E}(Y^2) \geq 2 \operatorname{E}(...
r_faszanatas's user avatar
2 votes
1 answer
144 views

Generalization of Komlós–Major–Tusnády Approximation

The Komlós–Major–Tusnády Approximation (see Wikipedia) considers the sum of uniform variables in $(0,1)$. There are also version where instead the sum of equiprobable $0/1$ variables is used ($p=1/2$)....
user35154's user avatar
4 votes
1 answer
385 views

Expectation of exponential of a function of independent Rademacher r.v.'s involving the error function

Let $Z,Z'\in\{-1,1\}^n$ be two independent vectors of i.i.d. Rademacher r.v.'s, where $1\leq n \leq d$ are two integers ($d\gg 1$). I am trying to get an upper bound on $$ \mathbb{E}_{ZZ'}\left[ \exp\...
Clement C.'s user avatar
  • 1,342
6 votes
1 answer
1k views

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?...
phybrain's user avatar
  • 103
4 votes
0 answers
146 views

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 ...
zxzx179's user avatar
  • 205
3 votes
2 answers
259 views

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, ...
Bianca's user avatar
  • 41
5 votes
3 answers
4k views

Distribution of the individual coordinates of a uniform random vector on a high-dimensional sphere

Let $X=(X_1,\ldots,X_n)$ be a random vector uniformly distributed on the $n$-dimensional sphere of radius $R > 0$. Intuitively, i think that for large $p$ every coordinate $X_i$ is normally ...
dohmatob's user avatar
  • 6,716
0 votes
0 answers
84 views

If $(Y_n)_{n\in\mathbb N_0}$ and $(N_t)_{t\ge0}$ are stochastic processes, what is the filtration generated by $\left(Y_{N_t}\right)_{t\ge0}$?

Let $(\Omega,\mathcal A)$ and $(E,\mathcal E)$ be measurable spaces $(Y_n)_{n\in\mathbb N_0}$ be a $(E,\mathcal E)$-valued stochastic process on $(\Omega,\mathcal A)$ $(N_t)_{t\ge0}$ be a $\mathbb ...
0xbadf00d's user avatar
  • 161
11 votes
1 answer
1k views

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 ...
Xiao's user avatar
  • 425
1 vote
1 answer
144 views

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 ...
Kazuki OKAMURA's user avatar
1 vote
0 answers
134 views

Construction of Feller's pseudo-poisson process

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(E,\mathcal E)$ be a measurable space $(Y_n)_{n\in\mathbb N_0}$ be a $(E,\mathcal E)$-valued time-homogeneous Markov chain on $(\...
0xbadf00d's user avatar
  • 161
5 votes
4 answers
362 views

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. ...
Sam Spiro's user avatar
  • 470
-2 votes
1 answer
47 views

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 ...
Adrian Radillo's user avatar
3 votes
0 answers
162 views

Asymptotic behaviour of principal eigenfunctions and large deviations

Dear Math Overflowers, I am currently interested in a particular problem involving Large Deviations. I am only going to talk about the PDE side of the problem, but I'll be happy to provide more ...
leo monsaingeon's user avatar
2 votes
0 answers
214 views

Random walk and comparing sums of Exponential random variables

Let $\sigma$ be the time a nearest neighbor random walk started at 1 that has probability $p>1/2$ of moving left reaches $0$. Let $\sigma'$ be an independent copy of $\sigma$. Let $(X_k)_1^\infty$ ...
Matthew Junge's user avatar
5 votes
1 answer
911 views

Largest eigenvalues of a (random) correlation matrix?

I am recently studying on eigenvalues of a (random) correltion matrix. For a $N\times N$ correlation matrix (with a given meaning of randomness), its (1st, 2nd, etc.) eigenvalues have some ...
Jiyuan Zhang's user avatar
2 votes
0 answers
53 views

Sufficient condition for square root fluctuations of an ergodic sequence

Suppose I have a random sequence $\mathbf{X}=\{X_n\}_{n\in\mathbb{Z}}\subset \mathbb{R}^{\mathbb{Z}}$ that is ergodic with respect to translations. I am interested in a sufficient condition on $\...
Ofer's user avatar
  • 83
0 votes
1 answer
73 views

Expected sum of chosen coordinates in a random subset of a Hamming hypercube

Let $S$ = $\{v_1, v_2, ..., v_n\}$ denote a random subset of a Hamming hypercube of dimension $d$, where $n = |S|$ and $n \leq 2^d$. If $v_i$ = $\langle x^i_1x^i_2... x^i_d\rangle$ for all $i \in [1,n]...
kevin's user avatar
  • 11
6 votes
2 answers
261 views

Nonlinear boolean functions

Let $\mathbb{F}_2=\{0,1\}$ be the field with two elements. I wonder if there is any known algorithm/construction that, given any $n\geq 1$, returns a boolean function $f:\mathbb{F}^n_2\rightarrow \...
Michele's user avatar
  • 313
2 votes
2 answers
2k views

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 ...
Student's user avatar
  • 653
1 vote
1 answer
136 views

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?
Varidhi Shayana's user avatar
6 votes
2 answers
551 views

Probability of at least two of $n$ independent events occurring subject to some conditions

Given a set of independent Bernoulli random variables $\{x_1, \dots, x_n\}$, let $p = \sum_{0<i\leq n}\Pr[x_i = 1]$ and $X=\sum_{0<i\leq n} x_i$. We know that for any $i$, we have $\Pr[x_i = 1]\...
Melika's user avatar
  • 189
7 votes
1 answer
245 views

Collecting proofs of the birth of the giant component

I want to collect different proofs of Erdös-Rényi result on the double jump of the largest connected component on $G(n,p)$ (or in $G(n,M)$. I know the original proof of Erdös-Rényi, the proof that ...
Johnny Cage's user avatar
  • 1,543
1 vote
1 answer
89 views

gaussian isoperimetric result for minimal measure under translation

Consider two spherical Gaussian distributions in $\mathbb{R}^n$, $A = \mathcal{N}(x, I)$ and $B=\mathcal{N}(y, I)$ where the difference in means is $\delta = y - x$. Let $S \subset \mathbb{R}^n$ be a ...
Jeremy's user avatar
  • 133
4 votes
0 answers
235 views

What happens in the martingale CLT if I norm by the conditional variance instead?

TLDR: I'm a statistician (bear with me!) trying to use the martingale CLT but I only can estimate the conditional variance instead of the unconditional one. Can I do anything to get a CLT with norming ...
gogurt's user avatar
  • 91
2 votes
1 answer
379 views

References for Hellinger distance/affinity involving mixture distributions

For two continuous probability distributions $F,G$ and their densities, $f,g$, the (squared) Hellinger distance/affinity is given by $d^2_H(F,G)=1-\int_{\mathbb{R}} \sqrt{fg}~dx$. Suppose that $f,g$ ...
stats134711's user avatar
2 votes
1 answer
263 views

The effect of random projections on matrices

Let $A\in\mathbb{R}^{n\times n}$ be a given normal matrix, i.e. $A^TA=AA^T$. Let $P_s\in\mathbb{R}^n$ be a random projection matrix to an $s$-dimensional subspace in $\mathbb{R}^n$. Suppose $\frac{A+...
neverevernever's user avatar
3 votes
1 answer
234 views

Concentration of a modified random walk

Let $\varepsilon$ be a number in $(0, 1)$, consider the following random walk on the real line $X^{(0)}, X^{(1)}, \dots$, where $X^{(0)}=0$ If $X^{(t)} > 0$, then with probability $.5$, $X^{(t+1)...
Xi Wu's user avatar
  • 143
8 votes
3 answers
252 views

Random reflections unexpectedly produce banded distributions

Start with $p_1$ a random point on the origin-centered unit circle $C$. At step $i$, select a random point $q_i$ on $C$, and a random mirror line $M_i$ through $q_i$, and reflect $p_i$ in $M_i$ to ...
Joseph O'Rourke's user avatar
2 votes
0 answers
130 views

Open problems in Monte Carlo Simulation [closed]

I want to know some open problems in Monte Carlo Simulation, which is being studied or in a stalemate. Could you please give me some advice? Thanks a alot
Phát Đạt Nguyễn's user avatar
5 votes
1 answer
397 views

Random pairs of commuting permutations

Let $\Omega_n \subseteq \mathrm{Sym}(n)^4$ be the set of all $4$-tuples $(\sigma_1,\sigma_2,\tau_1,\tau_2)$ of permutations of $\{1,\ldots,n\}$ such that $\sigma_j \tau_k = \tau_k \sigma_j$ for each ...
burtonpeterj's user avatar
  • 1,689
2 votes
1 answer
282 views

Gaussian sum VS Brownian motion

Given independent Gaussian $d$ dimensional vectors $G_i$, Let $ \sigma^2_n=\mathbb{E}(\sum_{i \le n} G_i) \cdot (\sum_{i \le n} G_i)^T$. $||\sigma_n^2||$ is norm of $\sigma_n^2$. Is there a $d$-...
jason's user avatar
  • 553
4 votes
2 answers
392 views

Effect of perturbing the atoms of a measure on the Wasserstein distance

Let $(X,d)$ be a metric space, $x_1,\ldots,x_N\in X$ and $x_1',\ldots,x_N'\in X$ be atoms, and $G=\sum_{i=1}^Np_i\delta_{x_i}$, $G'=\sum_{i=1}^Np_i'\delta_{x_i}$, and $G''=\sum_{i=1}^Np_i'\delta_{x_i'}...
JohnA's user avatar
  • 680
6 votes
3 answers
466 views

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 ...
Dominic van der Zypen's user avatar
2 votes
1 answer
65 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 $...
Viktor B's user avatar
  • 704
2 votes
2 answers
126 views

Spectral decomposition of a combinatorial matrix/Random walks on $s$-sets

$\newcommand{\Z}{\mathbb{Z}} \newcommand{\J}{\mathcal{J}} \newcommand{\la}{\lambda} \newcommand{\1}{\mathbf{1}} \newcommand{\R}{\mathbb{R}}$ Take any $n\in[3;\infty]$. Here and in what follows, $[k;\...
Iosif Pinelis's user avatar
5 votes
0 answers
1k views

Asymptotic behavior of row sums in 2-d array of random variables

Set-up. Let $f : \mathbb{N} \to \mathbb{N}$ be increasing. For each $m \in [0,1]$, consider an infinite two-dimensional array of random variables, where row $n$ has $f(n)$ variables: $B^m_{1,1}$ $B^...
cosmo-grant's user avatar
3 votes
0 answers
321 views

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 ...
Nick Broderick's user avatar
2 votes
0 answers
116 views

An Incorrect Construction of the Ito Integral

Let $B_t$ be a Brownian motion defined on the interval $[0,T]$, with underlying (filtered) probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\},\mathbb{P})$. Call a function $f:[0,T]\times\Omega\...
Adam Quinn Jaffe's user avatar
3 votes
1 answer
331 views

Eigenvalues of random matrix conditional on positive definiteness

Consider the Gaussian Orthogonal Ensemble, considered as a probability measure $\mu$ on the space of real symmetric matrices. Let $\mu|PD$ denote this measure conditioned on the event that the matrix ...
Simon Segert's user avatar

1
71 72
73
74 75
173