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.
8,631
questions
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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 $n$ ...
4
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0
answers
223
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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;
$...
4
votes
0
answers
768
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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\\...
4
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0
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66
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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:
$$(1/c)|I|^{a}\leq\mu(I)\leq c|I|^{b},$$
$$\sup_{x\in [0,1]}\frac{\mu(B_{2r}(x))}{\mu(B_{r}(x))^{1-...
4
votes
0
answers
318
views
Distribution of min/max row sum of matrix with i.i.d. uniform random variables
Given a $n\times n$ symmetric random matrix such that
all diagonal elements are all fixed as $1$.
all elements in upper triangle (excluding the diagonal) are i.i.d. uniform random variables ...
4
votes
1
answer
119
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Closure of polynomials in $L^2_w$ with log-normal weight function
Consider the Hilbert space $L^2_w$ with scalar product $\langle f,g\rangle_w =\int_0^\infty f(x)g(x)w(x)dx$ where the weight $w$ is the density function of a log-normal distribution
$$ w(x)=\frac{1}{\...
4
votes
0
answers
2k
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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 ...
4
votes
0
answers
149
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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\...
4
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0
answers
545
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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 ...
4
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0
answers
90
views
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 ...
4
votes
0
answers
181
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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 ...
4
votes
0
answers
178
views
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 ...
4
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0
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727
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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 ...
4
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0
answers
505
views
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 ...
4
votes
0
answers
338
views
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_{...
4
votes
0
answers
91
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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
\...
4
votes
0
answers
223
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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 ...
4
votes
0
answers
127
views
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}\...
4
votes
0
answers
51
views
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 ...
4
votes
0
answers
97
views
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 $...
4
votes
0
answers
213
views
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 ...
4
votes
0
answers
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 $\...
4
votes
1
answer
366
views
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 $m$ ...
4
votes
0
answers
56
views
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-...
4
votes
0
answers
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-...
4
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0
answers
238
views
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^...
4
votes
0
answers
81
views
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
$$\max \lbrace{\sum_{1 \leq i,j \leq n} \epsilon_i \;...
4
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0
answers
136
views
The asymptotic behavior of the ratio between the largest two of $n$ i.i.d. chi-square random variables
My question is about the asymptotic behavior of the ratio between the largest and second largest values of $n$ independent chi-square random variables.
Let $X_1, \ldots, X_n$ be $n$ independent and ...
4
votes
0
answers
517
views
Inequality for Conditional Variance
Let $X,Y$ be real random variables with joint density $p(x,y)$. Let the variance of $X, Y$ be $\operatorname{Var}(X) = \operatorname{Var} (Y)=\sigma^2$. The conditional variance of $X$ for a given $Y=...
4
votes
0
answers
95
views
Approximating martingales given marginal distributions
Let $(\mu_0,\mu_1)$ be a vector of probability measures on $\mathbb R$ that are of finite first moment, i.e.
$$\int_{\mathbb{R}}|x|\mu_i(dx)~<~+\infty \mbox{ for } i=0,1$$
and increasing in ...
4
votes
0
answers
71
views
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}: \...
4
votes
0
answers
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 ...
4
votes
0
answers
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 \...
4
votes
0
answers
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}\...
4
votes
0
answers
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^\...
4
votes
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 ...
4
votes
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 ...
4
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0
answers
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$-...
4
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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 ...
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{...
4
votes
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}\...
4
votes
0
answers
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)$ ...
4
votes
0
answers
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 ...
4
votes
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 ...
4
votes
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 ...
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 ...
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 ...
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 ...
4
votes
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 $i$.
...
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 ...