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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Random packing density of cylinders in a volume

I am trying to calculate the packing density of cylindrical bottles in a box, assuming that the bottles are randomly dumped in the box. I have read on the packing density of spheres here https://en....
David G's user avatar
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Bound that random walk stays within with constant probability?

For one-dimensional random walk, it is well-known that if the walk goes for $n$ steps, with constant probability it ends within $\pm\sqrt{n}$. What is the bound, in terms of $n$, such that if the ...
pi66's user avatar
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Deduce average order of $\phi(n)/n$ from probability that two integers are coprime

I've seen proofs of the fact that the probability of two random integers being coprime is $\frac{6}{\pi^2}$ (all of them leading to a use of the Riemann Zeta function and the Basel problem). In ...
Sir Sanderson's user avatar
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Expectation of Gaussian random vector & arbitrary function thereof?

I saw in a paper (https://www.princeton.edu/~wbialek/rome/refs/bialek+ruyter_05.pdf Eq.37) the following identity: where the <.> operator refers to a population average. No source or ...
DankMasterDan's user avatar
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When does the cumulative distribution function solve the Kolmogorov backward equation?

For a diffusion $X$ define the cumulative distribution function for $X_T$ started with $X_t=x$: $$u(t,x):=E^{t,x}(1_{X_T\ge y})$$ Under what conditions does $u$ solve $X$'s Kolmogorov backward ...
JSG's user avatar
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Expected number of non-empty regions

Consider $d$ dimensional space cut by $n$ hyperplanes in general position, each one of which goes through the origin. The number of distinct regions created is known to be: $$2\sum_{i=0}^{d-1} {n -1 \...
Simd's user avatar
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GOE convergence

As is well-known (at least in some circles), eigenvalue spacing distribution for large symmetric matrices converges as size goes to infinity (see this question for more background). The question is: ...
Igor Rivin's user avatar
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Probability that a randomly filled Go board has a set of white stones connected through their von Neumann neighborhoods

I have an $N$ by $M$ grid (a Go board for example), where for every square in the grid, I place a white stone with probability $p$ and a black stone with probability $(1-p)$. We call two white stones ...
Roger S.'s user avatar
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Conditional geometric distributions

If $p<1$ and $X$ is a random variable distributed according to the geometric distribution $P(X = k) = p (1-p)^{k-1}$ for all $k\in \mathbb{N}$, then it is easy to show that $E(X) = \frac 1p$, $\...
Vaughn Climenhaga's user avatar
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Stochastic integrals as honest martingales -- comparison criterion

We have a given positive martingale $\rho_t$, with the dynamics: $$\textrm{d} \rho_t = \lambda_t \rho_t \textrm{d} W_t$$ where $W_t$ is a standard Brownian motion. Now we have a "dumped" process p_t: $...
Grzenio's user avatar
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Covariance sign

Hi, Is it true that $Cov[f(X),g(X)]>0$ where $X$ is a random variable of unbounded support and $f,g$ are two strictly increasing real functions? I think by Chebyschev integral inequality I must ...
quema's user avatar
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Why doesn't Stein effect happen for multinomial distributions?

(Medeen, et all, 1998)" show that Maximum Likelihood estimate is admissible for multinomial distribution under squared error. On other hand, James and Stein showed that arithmetic average is not an ...
Yaroslav Bulatov's user avatar
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2 answers
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Definition of weak conditional convergence of random variables

I am looking for a definition of conditional convergence. Suppose that $X_1, X_2, \dots, X_n$ are $\mathbb R$-valued random variables with finite second moments, and $W_1, W_2, \dots, W_n$ are iid $\...
Syd Amerikaner's user avatar
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An order statistics problem with some interesting geometry

Let $a_n$ be a given sequence of positive numbers, and $X_n$ a sequence of independent random variables with each $X_n$ uniformly distributed on $[0, a_n]$. Question: Let $N \geq 2$ be an arbitrary ...
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Grouping lists together in a proportional election: image of a Dirichlet distribution by the D'Hondt method of proportional allotment

A real-world motivation for this question is given below. But first let me recall what the Jefferson-D'Hondt “greatest divisors” method of proportional allotment (often used in electoral systems to ...
Gro-Tsen's user avatar
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Exponential of supremum of Brownian bridge on short time frame

For each $T > 0$, let $B^T$ be a Brownian bridge on $[0, T]$, conditioned to start and end at $0$. Question: Is it true that $\mathbb E[|\text{exp}\, (\sup_{0 \leq t \leq T} B^T_t) - 1|] \to 0$ as $...
Nate River's user avatar
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The number of $3$-CNF formulas in $n$-variables and the fraction of satisfiable ones

What is the number of $3$-CNF (conjunctive normal form) formulas with $n$ sentential variables and what is the fraction of satisfiable ones? I consider two formulas the same if they are syntactically ...
user1642683's user avatar
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2 answers
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For a SDE with smooth transition densities, if every point is "path-accessible", is every positive-measure set probabilistically accessible?

Suppose we have a $C^\infty$ manifold $M$ and $C^\infty$ vector fields $b,\sigma_1,\ldots,\sigma_k$ on $M$, and for convenience define the set of vector fields $$ \mathcal{S} = \{b,\sigma_1,-\sigma_1,\...
Julian Newman's user avatar
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Is the density of integers $n$ such that the finite sequence $(\omega(n-r)\omega(n+r))_{0\leq r\leq n-1}$ is surjective positive?

Let $\omega(m)$ be the number of prime factors of $m$ regardless of multiplicity. I'm interested in the behavior of the finite sequence $(\omega(n-r)\omega(n+r))_{0\leq r\leq n-1}$ for a given integer ...
Sylvain JULIEN's user avatar
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2 answers
370 views

Request for recommendation in probability and complex analysis

Could somebody kindly recommend to me some books which deal with the applications of the probabilistic method to problems in real and complex analysis or which consider probabilistic versions of some ...
AgnostMystic's user avatar
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Well-definedness of maximum likelihood estimation

Consider a family $\{\mu_\theta:\theta\in\Theta\}$ of probability measures on a measurable space $X$. Given $x\in X$, the maximum likelihood estimate is the value of $\theta$ which maximizes the ...
Quarto Bendir's user avatar
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What exactly is the relation between the Wiener process and Wiener measure?

The Wiener measure is (in the classical sense) a Gaussian measure on the Banach space $C[0,1]:=\{f:[0,1] \to \mathbb{R} \mid f\text{ is continuous and } f(0)=1\}$. The Wiener process is a stochastic ...
Isaac's user avatar
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Deriving an asymptotic statement from a recursion

The last few days I am trying my best to understand a part of a proof from these lecture notes on page 14: Picture of the relevant part The setting is percolation on a regular tree with degree $r$, $...
Testname420's user avatar
3 votes
1 answer
155 views

Probability permutation in turned to cycle

Let $M$ be a $0/1$ square matrix having one $1$ per row and column (permutation matrix). If you permute the columns and rows independently what is the probability resulting permutation matrix is a ...
Turbo's user avatar
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Existence of Gaussian random field with prescribed covariance

Suppose a function $G:\mathbb{R}^d\rightarrow\mathbb{R}$ is given. What are some necessary or sufficient conditions on $G$ for there to exist a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and ...
Cabbage's user avatar
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Request for books/articles on random polynomials

Can somebody kindly recommend me a couple of introductory books/articles on random polynomials with clear expositions of fundamental results (like the distribution of roots, expected number of real ...
AgnostMystic's user avatar
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1 answer
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Singular value decomposition of random rectangular matrices

Let $A$ be a $m\times n$ real matrix, whose entries are independent, identically distributed random variables, following standard normal distributions (mean zero and unit variance). What is the ...
valle's user avatar
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Gaussian concentration inequality

Recently I found a concentration inequality for infinite dimensional Gaussian r.v.s in this paper. Specifically, Lemma 4 on page 307 states (without a proof) that There exists a universal constant $...
d.k.o.'s user avatar
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2 answers
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Concentration of sum of concentrated random variables

I have a sum of positive random variables, they are not identically distributed, but even that case would be interesting. They are not necessarily independent, but I already have a concentration bound ...
user2316602's user avatar
3 votes
1 answer
247 views

Lipschitz functions that saturate the Lipschitz inequality on the average (part 1)

Consider a 1-Lipschitz function $f: \mathbb R^n \to \mathbb R$ satisfying the inequality \begin{align*} |f(x) - f(y)| \le \|x-y\|_2, \;\forall x,y \in \mathbb R^n. \end{align*} For $n \ge 2$, can we ...
passerby51's user avatar
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Pathwise stochastic integral as a linear operator on continuous functions

Let $B$ be a Brownian motion. Definining a pathwise stochastic integral $I(f):=\int f~dB$ for certain classes of deterministic functions is straightforward: For instance if $f=\sum_ic_i1\{[t_i,t_{i+1})...
user78370's user avatar
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1 answer
253 views

Functions $f$ such that $A$ and $f(A,B)$ are independent

Assume that $A$ and $B$ are two dependent random variables. Are there any results on functions $f$ such that $C =f(A, B)$ and $A$ are independent? For example, it can easily be shown that $A$ and $...
user avatar
3 votes
1 answer
309 views

Spectral radius of Markov averaging operator on graphs

The definition of Markov operator which I am familiar with: For a graph $G=(V,E)$, Markov's operator upon a function $\varphi:V\rightarrow\mathbb{C}$ , $\varphi\in L^2(G,\nu)$ ($\nu(v):=\deg(v)$) ...
user140823's user avatar
3 votes
2 answers
301 views

Lower bound Renyi divergence between two discrete probability distributions

I am trying to understand the proof of Lemma 1 in this paper (Section 9.2). The proof shows that given a discrete probability distribution $P=(p_1,p_2,...,p_k)$ where $p_1 \geq p_2 \geq ... \geq p_k$,...
Elwood Crandall's user avatar
3 votes
1 answer
315 views

Where to find the proof of this property?

I am doing some exercises in the analytic and there is a problem as following: ``Let $\{f_n\}_{n \in \mathbb{ N}}$'' to be a positive sequence such that: $\sum\limits_{n=1}^{+\infty} f_n = 1$. $\...
mathJuan's user avatar
  • 153
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2 answers
157 views

continuity/ measurablity of optimal transport

given polish space $(X,d)$, consider weak* topology of probability. optimal transport of probability $u,v$ is defined by $\pi(u,v)$ such that $\pi(u,v)$ minimizes: $\{\int d(x,y) d \pi(dx,dy): \pi \...
jason's user avatar
  • 553
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2 answers
335 views

Non-probabilist term for conditional expectation?

When writing an article I encounter what is essentially a conditional expectation - function defined on a bounded interval (not necessarily of unit length) with Lebesgue measure, but information about ...
Tommi's user avatar
  • 648
3 votes
2 answers
394 views

Central limit theorem for weak dependent bernoulli random variables

Suppose $\epsilon_1,\epsilon_2,...$ are i.i.d bounded random variables with compact support. Let $X_k=g_k(\epsilon_k,...,\epsilon_1)$ be Bernoulli random variables with the covariance between $X_i$ ...
joeyg's user avatar
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2 answers
303 views

Log-concavity of the maximum of gaussians

Let $Z_1,\ldots, Z_n$ be independent standard gaussian random variables. Is it true that $X=\max\{Z_1,\ldots,Z_n\}$ has a log-concave distribution function?
TOM's user avatar
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1 answer
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Is there a complete countable axiomatization of conditional independence? (Graphoids)

Note: A pointer to a reference, or a yes/no answer with a 1-2 sentence incomplete/non-rigorous justification would suffice for answers. I am just curious about whether the result is true; it is fairly ...
Chill2Macht's user avatar
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3 votes
2 answers
678 views

Concentration inequality for sum of iid random variables that involve KL distance

Conider $X \in \mathbb{R}^d$ and $Y \in \{0,1\}$, and a joint distribution $p_{XY}(x,y)$, and a set of $N$ i.i.d. samples $\{(X_i,Y_i)\}_{i=1}^{N}$. Define $p_{X0} = p_{XY}(x,0)$ and $p_{X1} = p_{XY}(...
Jeff's user avatar
  • 482
3 votes
2 answers
198 views

Percolation on finite irregular trees

Consider a rooted tree of height $h$, such that all the leaves are at last layer. We perform the following random process: each edge is deleted with probability $0.5$, and otherwise it is retained. We ...
Or Meir's user avatar
  • 419
3 votes
1 answer
153 views

Tight L2 bound on moments approximation and reference

Consider $f\in L^2(I)$, where $I$ is the unit interval and $L^2$ is w.r.t. Lebesgue measure, and consider an approximation of $f$ denoted by $\tilde{f}\in L^2$. The error in approximated the moments ...
Amir Sagiv's user avatar
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3 votes
2 answers
486 views

Does the hitting time of +1/-1 of a Brownian motion posess a density?

The law of the hitting time of a 1-dimensional Brownian motion $W$ is well known, but I can't find any information on the density of the hitting time of $|W|$. I define $T=\inf \{t>0,|W|(t)= 1\}$. ...
Adrien Laurent's user avatar
3 votes
3 answers
288 views

A question in central limit theorem

Suppose $\{X_n,n\ge1\}$ are independent r.v., $E(X_n)=0$, $\operatorname{Var} \left(X_n\right)=\sigma_n^2<\infty$. Set $S_n=\sum_{i=1}^nX_i$ and $s_n^2=\sum_{i=1}^n\sigma_i^2$, assume $$\frac{S_n}{...
J.Mike's user avatar
  • 141
3 votes
2 answers
673 views

Birthday inequality for non-uniform distributions for fixed collision probability (random allocation, collision probability)

Question: Consider a distribution $D$, and $n$ i.i.d. random variables $X_i$, all distributed according to $D$. Let $p^D_2:=\Pr[X_1=X_2]$. What is a lower bound for $p^D_n:=\Pr[\exists i\neq j. X_i=...
Dominique Unruh's user avatar
3 votes
2 answers
417 views

Exchangeability and conditional expectation

The following problem arises in my research of interacting particle systems: Suppose that $X_1, \ldots, X_n$ are exchangeable real-valued random variables. Let $f,g: \mathbb{R} \to \mathbb{R}$ be ...
Richard's user avatar
  • 357
3 votes
1 answer
853 views

Cumulative integral of the Marchenko-Pastur density for Wishart eigenvalues

I can't find any work on the cumulative density function of the well-known Marchenko-Pastur density for the eigenvalues of a standard Wishart Matrix as its dimension goes to $\infty$, i.e. $$F(\beta)=...
blacklist's user avatar
3 votes
1 answer
254 views

spatial mean of log-normal random field

Let $X(x):= \exp(Y(x))$ for $x\in D\subseteq \mathbb{R}^d$ for a domain $D$, where $Y$ is a Gaussian random field with some smooth covariance function $c(\cdot,\cdot)\colon D\times D\to \mathbb{R}$. ...
user3095304's user avatar
3 votes
3 answers
413 views

A question about intuition of fluid limit in queuing system

This is a question about intuition in understanding the fluid limit queuing system. Assume we have a sequence of queuing systems $\{S^N\}_{N=1}^{\infty}$ with N servers and each server has unit ...
KevinKim's user avatar
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