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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Set version of ramsey type problem

For two sets of numbers $A,B$, write $A<B$ iff $\max A<\min B$. For a sequence of integers $a_0,\cdots,a_{n-1}>0$, let $Prop(a_0,\cdots,a_{n-1})$ denote the following proposition: Given $n$ ...
Jiayi Liu's user avatar
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Minimization over a convex function of equal vs unequal success probabilities of Bernoulli random variables

Let $U_1,U_2,\ldots,U_n$ be $n\geq 2$ mutually independent Bernoulli random variables. There are two cases of interest: $1.$ The random variables $U_1,U_2,\ldots,U_n$ are identically distributed; $...
Seyhmus Güngören's user avatar
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Total Variation distance of polynomials of Bernoulli R.V.s

Let $X_i, Y_i$ be i.i.d Bernoulli $0/1$ random variables with $\mathbb{E}[X_i] = p$ and $\mathbb{E}[Y_i] = q$. Let \begin{align*} X &= X_1 X_2 + Χ_2 Χ_3 + \ldots +X_{n-2} X_{n-1}+ X_{n-1} X_n\\...
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Tangent distribution for particular non-doubling measure (GMC)

The radon measure $\mu$ on [0,1] called GMC (Gaussian multiplicative chaos) satisfies the following: $$(1/c)|I|^{a}\leq\mu(I)\leq c|I|^{b},$$ $$\sup_{x\in [0,1]}\frac{\mu(B_{2r}(x))}{\mu(B_{r}(x))^{1-...
Thomas Kojar's user avatar
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318 views

Distribution of min/max row sum of matrix with i.i.d. uniform random variables

Given a $n\times n$ symmetric random matrix such that all diagonal elements are all fixed as $1$. all elements in upper triangle (excluding the diagonal) are i.i.d. uniform random variables ...
Tony's user avatar
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Closure of polynomials in $L^2_w$ with log-normal weight function

Consider the Hilbert space $L^2_w$ with scalar product $\langle f,g\rangle_w =\int_0^\infty f(x)g(x)w(x)dx$ where the weight $w$ is the density function of a log-normal distribution $$ w(x)=\frac{1}{\...
S. Willems's user avatar
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Does rate of convergence in probability come from a metric?

In general, when we talk about convergence of a sequence, we need a topological space. If we want to talk about a rate of convergence, we need to quantify how far away one element of the sequence is ...
Froomfondel's user avatar
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Dividing a finite arithmetic progression into two sets of same sum: always the same asymptotics?

This is inspired by the recent question How many solutions $\pm1\pm2\pm3…\pm n=0$. The oeis entries A063865 linked to this question and A292476/A156700 for the related one "How many solutions $\pm1\...
Wolfgang's user avatar
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Optimal transport between two distributions in a Markov chain

In a previous question, given an ergodic Markov chain, I'm interesting in sampling as short a path as possible with prescribed distributions for its endpoints. In a comment, I propose that the ...
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What is the entropy of binomial decay?

Let's play a game. I start with $N$ indistinguishable tokens, and I wait $T$ turns. Every turn, each token has probability $p$ of disappearing. I want an analytic formula for the entropy of this ...
Andrew's user avatar
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Distributions over permutation groups $\mathcal{S}_n$

Partly inspired by recent developments in enumeration of pattern avoiding permutations, which is known to be connected with Brownian excursions [Hoffman&Rizzolo]. The exciting milestone is the ...
Henry.L's user avatar
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Construct Lyapunov-Foster function given invariant distribution

Consider a discrete time Markov chain on a countable state space which is irreducible, aperiodic, and has a given invariant distribution $\pi$. Then the chain is necessarily positive recurrent and ...
Ian's user avatar
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KL Divergence - Convolution of distributions

Assume $P_1,P_2,P_3$ different to each other pmfs. We would like to find an upper bound for $D_{\text{KL}}(P_1\ast P_3||P_2 \ast P_3)$, where $D_{KL}$ is the Kullback-Leibler divergence and $*$ is ...
Cauchy's user avatar
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Sum of Binomial random variable CDF

Suppose there are two independent Binomial random variables $$ X\sim Binomial(n,p)\\ Y\sim Binomial(n,p+\delta) $$ where $\delta$ is considered to be fixed, and $p$ can vary in $(0,1-\delta)$. Now ...
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Lipschitz kernel

We consider the following probability measure on $\mathbb{R}^2$: $\mu = Leb\vert_{[0,1]} \times \delta_0$. Furthermore the following dilation, say $d$, is defined as $(x,0) \mapsto \frac{1}{2}(\delta_{...
tubmaster's user avatar
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Random variables whose expectations are cumulants

In my research I stumbled about the following class of random variables: Let $X_0,X_1,\dots$ be random variables on a common probability space with finite moments of all orders. We then define \...
Julian's user avatar
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Good introduction to Benjamini- Schramm limits [closed]

So I was wondering if someone might be able to suggest a good intro paper/ article for getting a feel for Benjamini- Schramm limits as well as getting a sense of the kinds of results that people have ...
David Pechersky's user avatar
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Is there an example that both Berry-Essen bound and DKW bound are attained?

The Berry-Essen bound stated that $$\sup _{{x\in {\mathbb R}}}\left|\widehat{F_{n}(x)}-\Phi (x)\right|\leq C_{0}\cdot \psi _{0}$$ where $\psi _{0}(n)={\Big (}{\textstyle \sum \limits _{{i=1}}^{n}\...
Henry.L's user avatar
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Expected value for difference of two dependent random subset

Let $X=\{1,...,n\}$. I select $m$ values uniformly, on at a time, with replacement from $X$. Let $A$ denotes this random subset and $|A|$ be the random variable for the number of distinct elements in ...
Sam's user avatar
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Does a non-exchangeable empirical reverse-martingale exist?

Consider a possible finite sequence $\xi_1,\xi_2,\dots$ of random variables and consider the measure-valued empirical process $$\eta_n=\frac{\sum_{i=1}^n\delta_{\xi_i}}{n},\:\:\: n=1,2,\dots$$ Assume $...
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Game theory of writing multiple choice tests

Here is a model which seems pretty close to my experience of writing multiple choice tests. Let's view the answer $t$ to each question as a binary string in $S:=\{ 0,1 \}^k$, all equally likely. The ...
David E Speyer's user avatar
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153 views

Concentration Inequality for Score Functions of Exponential Familty

Let $p$ be the density of a continuous one-parameter exponential family distribution on $\mathbb{R}$. We assume that $$p(x) = c(x)\cdot \exp\bigl [ x \cdot \theta - b(\theta ) \bigr ], $$ where $\...
Steve's user avatar
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Variance of load in maximally loaded bin, if $m$ balls are thrown into $n$ bins

In the paper "Balls and bins a simple and tight analysis" by Raab and Steger, available here strong upper and lower bounds are proved about the number $M$ of balls in a maximally loaded bin when $m$ ...
kodlu's user avatar
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Is there an equivalent line time-invariant system for a linear time-varying system with specific properties? [closed]

Given a discrete-time linear time-varying system (LTV) $$x(k+1) = A(k) x(k) + B(k) u(k)$$ where $A(k)$ and $B(k)$ are generated by a stationary random process. Is there an equivalent linear time-...
Ed Tate's user avatar
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405 views

Definition of the Stratonovich integral in Hilbert spaces

Let $T>0$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathcal F=(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration on $(\Omega,\mathcal A,\operatorname P)$ $B$ be a (standard, real-...
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Algebras and $\sigma$-algebras associated to random variables

Let $\{v_\lambda:~\lambda\in\Lambda\}$ be a family of real-valued random variables on a (complete) probability space $(\Omega, \sigma, \mathbb{P})$. Assume the variables lie in $\bigcap_{p=1}^\infty L^...
Ollie's user avatar
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The max of a random sum, SK model

Let $(N_{i,j}, i,j \in \mathbb N)$ be independent (standard) Gaussian random variables. What is known/ what is conjectured /what can we say about $$\max \lbrace{\sum_{1 \leq i,j \leq n} \epsilon_i \;...
Olivier's user avatar
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136 views

The asymptotic behavior of the ratio between the largest two of $n$ i.i.d. chi-square random variables

My question is about the asymptotic behavior of the ratio between the largest and second largest values of $n$ independent chi-square random variables. Let $X_1, \ldots, X_n$ be $n$ independent and ...
Steve's user avatar
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517 views

Inequality for Conditional Variance

Let $X,Y$ be real random variables with joint density $p(x,y)$. Let the variance of $X, Y$ be $\operatorname{Var}(X) = \operatorname{Var} (Y)=\sigma^2$. The conditional variance of $X$ for a given $Y=...
Vivek Bagaria's user avatar
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95 views

Approximating martingales given marginal distributions

Let $(\mu_0,\mu_1)$ be a vector of probability measures on $\mathbb R$ that are of finite first moment, i.e. $$\int_{\mathbb{R}}|x|\mu_i(dx)~<~+\infty \mbox{ for } i=0,1$$ and increasing in ...
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Regularity Conditions for L1 convergence of maximum likelihood estimators

Let $X_1,\ldots, X_n$ be i.i.d. observations from a family of pdf or pmf $\{f_{\theta}: \theta \in \Theta \}$. We know that there are sufficient regularity conditions on the family $\{f_{\theta}: \...
sde's user avatar
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189 views

Remaining models conjectured to converge to SLE(6) or CLE(6)

I am wondering which models are conjectured (eg. numerically) to converge to SLE(6) (Schramm-Loewner evolution with $\kappa=6$) or CLE(6) (conformal loop ensemble). I am searching for a research topic ...
user133100's user avatar
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178 views

Volume difference in random approximation of polytope

The following easily stated problem has arisen in my research. However, it's outside my field, and I'm unfamiliar with the literature. I would greatly appreciate any references. Let $K\subset \...
Paul Constantine's user avatar
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137 views

Level sets of function of inner products of vectors on hypercube

Let $H = \{ 0, 1\}^d$ be the $d$-th Cartesian product of $\{0, 1\}$ in $\mathbb{R}^d$. Suppose $v_1, \ldots, v_k$ are $k$ vectors in $H$ in general position. We define function $F \colon H^{k}\...
Steve's user avatar
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76 views

How well does an estimator perform on another dataset?

Suppose $X \sim N(0, \Sigma)$ is a $d$-dimensional Gaussian random vector. And we have $2n$ $i.i.d$ sample $X_1, \ldots, X_{n}, \ldots, X_{2n}$. Let $\hat{\Sigma}_1 = \frac{1}{n}\sum_{i=1}^nX_i X_i^\...
Wuchen's user avatar
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0 answers
128 views

Maximum Likelihood and De Finetti's Theorem

I have a question about whether it is possible to use De Finetti's representation theorem for maximum likelihood estimation. De Finnetti's theorem states that for any exchangable infinite sequence of ...
Asterix's user avatar
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0 answers
106 views

Random polyominoes containing $2\times2$ squares

The construction quoted in the question "How to sample a uniform random polyomino?" claims to produce a "uniform random polyomino". But apart from the mentioned possibility of getting stuck, it also ...
Wolfgang's user avatar
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248 views

Concentration of infinite-dimensional Gaussian measure

I have the question about finding the subspace of concentration of a Gaussian Measure. More precisely: $\textbf{Question:}$ Assume we have a separable Hilbert space $\ell_2$ with Borel $\sigma$-...
Fedor Goncharov's user avatar
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0 answers
120 views

Short time asymptotics for Brownian motion on a compact manifold

Consider a compact Riemannian manifold $(M, g)$. Choose a ball $B(p, r)$ inside $M$, and a quasi-isometric ball $B(q, s)$ in $\mathbb{R}^n$, in the image of a coordinate chart containing $B(p, r)$ (in ...
user94063's user avatar
4 votes
0 answers
387 views

concentration of functions of Gaussian processes

Let $\mathcal{C}\in\mathbb{R}^n$ be a subset of the unit ball. Also let $\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_m\in\mathbb{R}^n$ be i.i.d. random Gaussian vectors $\mathcal{N}(\mathbf{0},\mathbf{...
mohi's user avatar
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0 answers
229 views

Self-adjusting random walk

Let $X_t$ be a random process such that \begin{eqnarray} X_1 &=& 0\\ X_t &=& X_{t-1} + \left\{\begin{array}{ll} A_t, & X_{t-1} \geq 0\\ B_t, & X_{t-1} < 0 \end{array}\...
Kasper's user avatar
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551 views

An inequality involving conditional variance and its connection to information theory

Given absolutely continuous random variables $(X, Y)$ with joint distribution $P_{XY}$, we construct $Z:=\sqrt{\gamma} Y+N_\mathsf{G}$ where $N_\mathsf{G}\sim N(0, 1)$ and is independent of $(X,Y)$ ...
math-Student's user avatar
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873 views

Is a local martingale with constant expectation necessarily a martingale?

Suppose $X\in \mathbb R$ is a weak solution to the SDE $dX_t = \sigma(X_t)dW_t$, in which $W$ is a one-dimensional Brownian motion, and $\sigma$ is Borel measurable so that a weak solution exists and ...
epsilon's user avatar
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0 answers
146 views

What is the influence of unreliable comparisons on the results of sorting

Considering sorting algorithms based solely on binary comparisons of the elements to be sorted(algorithms such as insertion sort, selection sort, quicksort, and so on), what problems do we face when ...
Jessy's user avatar
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0 answers
165 views

Is there any probabilistic characterization for generalized solvable groups?

References: This question is inspired by a conjecture of Alon Amit that is solved by Miklós Abért, Nikolay Nikolov and Dan Segal in the following papers: (1) On the probability of satisfying a word in ...
user avatar
4 votes
0 answers
452 views

The distribution of the elements of an eigenvector of random matrices

Suppose a random matrix $A$ with its elements following Gaussian distribution with non-zero mean. We know that the eigenvalues of $A$ have two patches: one is at the real axis that is far away from ...
Zedong Bi's user avatar
4 votes
0 answers
130 views

regularity of zero point

We consider 1-d process $X$ $$ X(t) = b t + J_{t} + M_{t}$$ where $b$ is constant, $M$ is a continuous martingale process with $M(0) = 0$, and $J$ is a symmestric $\alpha$-stable process with its ...
kenneth's user avatar
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4 votes
0 answers
94 views

Finding closest set of K disjoint hyperspheres to a point in $\mathbb{R}^n$ with uniform radius

I am interested in the following problem: in $\mathbb{R}^n$, we have $N$ overlapping hyperspheres all with the same radius. Given a point $p$ in $\mathbb{R}^n$, the objective is to find the $K$ non ...
eagle34's user avatar
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0 answers
202 views

Dimension reduction for low-order moments of Rademacher-weighted sums of vectors

Let $x_1,\dots,x_n$ be vectors in a Euclidean space $H$. Let $\varepsilon_1,\dots,\varepsilon_n$ be independent Rademacher random variables (r.v.'s), so that $P(\varepsilon_i=\pm1)=1/2$ for all $i$. ...
Iosif Pinelis's user avatar
4 votes
0 answers
136 views

Explicit formula for the moment problem with Carleman's conditions

Suppose that $\mu_j$ are real numbers obeying the Carleman condition: $\sum_{j=0}^\infty 1/\mu_{2j}^{2j}<\infty$. Then it is well-known that in case the $\mu_j$ are positive definite, there is a ...
Jeff's user avatar
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