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.
8,630
questions
3
votes
2
answers
9k
views
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....
3
votes
2
answers
1k
views
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 ...
3
votes
2
answers
248
views
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 ...
3
votes
2
answers
273
views
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 ...
3
votes
1
answer
528
views
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 ...
3
votes
2
answers
301
views
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:
$$2\sum_{i=0}^{d-1} {n -1 \...
3
votes
1
answer
127
views
GOE convergence
As is well-known (at least in some circles), eigenvalue spacing distribution for large symmetric matrices converges as size goes to infinity (see this question for more background). The question is: ...
3
votes
1
answer
257
views
Probability that a randomly filled Go board has a set of white stones connected through their von Neumann neighborhoods
I have an $N$ by $M$ grid (a Go board for example), where for every square in the grid, I place a white stone with probability $p$ and a black stone with probability $(1-p)$. We call two white stones ...
3
votes
3
answers
2k
views
Conditional geometric distributions
If $p<1$ and $X$ is a random variable distributed according to the geometric distribution $P(X = k) = p (1-p)^{k-1}$ for all $k\in \mathbb{N}$, then it is easy to show that $E(X) = \frac 1p$, $\...
3
votes
1
answer
536
views
Stochastic integrals as honest martingales -- comparison criterion
We have a given positive martingale $\rho_t$, with the dynamics:
$$\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:
$...
3
votes
3
answers
2k
views
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 ...
3
votes
1
answer
558
views
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 ...
3
votes
2
answers
244
views
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 $\...
3
votes
1
answer
378
views
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 ...
3
votes
2
answers
244
views
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 ...
3
votes
1
answer
146
views
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 $...
3
votes
1
answer
239
views
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 ...
3
votes
2
answers
231
views
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,\...
3
votes
1
answer
136
views
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 ...
3
votes
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 ...
3
votes
1
answer
354
views
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 ...
3
votes
2
answers
2k
views
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 ...
3
votes
2
answers
229
views
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 $r$, $...
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 ...
3
votes
1
answer
120
views
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 ...
3
votes
2
answers
195
views
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 ...
3
votes
1
answer
2k
views
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 ...
3
votes
1
answer
1k
views
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 $...
3
votes
2
answers
1k
views
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 ...
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 ...
3
votes
1
answer
463
views
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})...
3
votes
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 $...
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
$\varphi:V\rightarrow\mathbb{C}$ , $\varphi\in L^2(G,\nu)$ ($\nu(v):=\deg(v)$) ...
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 $p_1 \geq p_2 \geq ... \geq p_k$,...
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:
$\sum\limits_{n=1}^{+\infty} f_n = 1$.
$\...
3
votes
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 \...
3
votes
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 ...
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 $X_i$ ...
3
votes
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?
3
votes
1
answer
309
views
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 ...
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}(...
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 ...
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 ...
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 $T=\inf \{t>0,|W|(t)= 1\}$. ...
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
$$\frac{S_n}{...
3
votes
2
answers
673
views
Birthday inequality for non-uniform distributions for fixed collision probability (random allocation, collision probability)
Question: Consider a distribution $D$, and $n$ i.i.d. random variables $X_i$, all distributed according to $D$. Let $p^D_2:=\Pr[X_1=X_2]$. What is a lower bound for $p^D_n:=\Pr[\exists i\neq j. X_i=...
3
votes
2
answers
417
views
Exchangeability and conditional expectation
The following problem arises in my research of interacting particle systems:
Suppose that $X_1, \ldots, X_n$ are exchangeable real-valued random variables. Let $f,g: \mathbb{R} \to \mathbb{R}$ be ...
3
votes
1
answer
853
views
Cumulative integral of the Marchenko-Pastur density for Wishart eigenvalues
I can't find any work on the cumulative density function of the well-known Marchenko-Pastur density for the eigenvalues of a standard Wishart Matrix as its dimension goes to $\infty$, i.e.
$$F(\beta)=...
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 $c(\cdot,\cdot)\colon D\times D\to \mathbb{R}$. ...
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 ...