Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

learn more… | top users | synonyms (1)

5
votes
0answers
120 views

Elementary function relative to erf

The modified Bessel function of the 1st kind $I_0$ is defined by $$ I_0(z)=\frac1\pi\int_0^{2\pi}e^{z\cos\theta}\,d\theta $$ and arises, among other places, in the probability density function of a ...
11
votes
0answers
377 views

Transitivity of balanced mass transport in Z

Given two atomic measures $\mu$ and $\nu$ on $\mathbb{Z}$, write $\mu \sim \nu$ iff there exist countable decompositions $\mu = \mu_1 + \mu_2 + \cdots$ and $\nu = \nu_1 + \nu_2 + \cdots$ along with ...
2
votes
1answer
101 views

Variant of Skorokhod's theorem

Consider the following situation: $S, T$ are standard Borel spaces (say $S = [0,1]^k$, $T = [0,1]$ if it is helpful). There is a a random variable $\zeta: \Omega \to S$. $f_n(\zeta) \to^d \eta$, ...
2
votes
0answers
53 views

Smallest Singular Value of a Random Matrix with Dependent Entries

Overview I am trying to bound from below the smallest singular value $\sigma_{n}$ of a sequence of symmetric $n$ by $n$ random matrices $M_{n}$ with dependant entries. In particular, I would like to ...
2
votes
1answer
145 views

Weak convergence of probability measures on weak versus strong dual

The space of temperate distributions $S'(\mathbb{R}^d)$ is often equipped with the weak-$\ast$ or with the strong topology. When defining the notion of a probability measure on $S'(\mathbb{R}^d)$, ...
5
votes
0answers
229 views

Squaring random Schwartz distributions

Let $\mu$ denote the centered Gaussian measure on $S'(\mathbb{R}^d)$ with covariance $$ \mathbb{E} [\phi(f)\phi(g)]=\int_{\mathbb{R}^d} \frac{\overline{\widehat{f}(\xi)} ...
6
votes
2answers
151 views

Tail sigma-algebra of a branching random walk

I am looking for any known results about the tail sigma-algebra of a branching random walk. To be specific, let $T$ be the nodes of an infinite binary tree rooted at $r \in T$. Let $\{X_t\})_{t \in ...
4
votes
1answer
118 views

Estimating the distribution of minimal hamming distances within a set of strings?

Is their an efficient mathematical way to estimate the distribution of minimal hamming distances for a set of random strings of length 8 over a 4-letter alphabet? E.g. given a set of 100-10,000 ...
13
votes
2answers
415 views

The power of two random choices with pairwise independence

Throw $n$ balls into $n$ bins, and let $X_n$ be the max load. That is the number of balls in the fullest bin. It is known that if the balls are thrown uniformly and independently at random then ...
0
votes
1answer
113 views

Probability Content of a random ball in R^n

As a follow up to this question, concerning this paper: Given random variables $X_1,\ldots,X_N,X_q:\Omega\rightarrow\mathbb{R}^d$, where $X_1,\ldots,X_N$ are independent and identical distributed. ...
8
votes
1answer
304 views

Transitive closure of balanced mass transport in Z (move to close)

Given two atomic measures $\mu$ and $\nu$ on $\mathbb{Z}$, write $\mu \sim \nu$ iff there exist countable decompositions $\mu = \mu_1 + \mu_2 + \cdots$ and $\nu = \nu_1 + \nu_2 + \cdots$ along with ...
12
votes
1answer
397 views

Natural probability on integers

This is a follow-up to this classical question asked recently here: we know (e.g. using the second Borel-Cantelli Lemma) that no probability measure on $\mathbb{Z}$ has the property that $n\mathbb{Z}$ ...
2
votes
2answers
91 views

Moment problem for discrete distributions

Let $x_1, \dots, x_N \in \mathbb R$ and consider the discrete distribution $\mu := \frac{1}{N} \sum_{i=1}^N \delta_{x_i}$, where $\delta_x$ denotes the Dirac measure, i.e. for any measurable set $B ...
16
votes
3answers
656 views

Existence of a “quasi-uniform” probablility distribution on $\mathbb{Z}$

Does there exist a probability distribution on $\mathbb{Z}$ such that for every integer $n\geq 1$, the probability that a random integer $x$ is divisible by $n$ equals $1/n$? Henry Cohn has an ...
-2
votes
1answer
116 views

If $(X_n+Y_n)$ has bounded variance, is the same true for $(X_n)$ and $(Y_n)$? [closed]

let $(X_n)$ and $(Y_n)$ be two sequences of random variables defined on the same probability space such that the variance of all components $X_n$, $Y_n$ is finite and the sequence of variances of ...
1
vote
0answers
67 views

Does the expected spreading of sample paths imply increase in variance?

Consider a sample-continuous stochastic process $\left\{ X_t \right\}_{t \in T}$ s.t. each $X_t$ is real-valued and $$\int_\Omega | X_t(\omega) | ^p \, \mathrm{d} P(\omega)< \infty$$ for all $1 ...
1
vote
1answer
98 views

Projective family of probability spaces

This is a crosspost of this question from MSE. I'm confused about the definition of a projective family of probability spaces $(S_t,\mathscr S _t,\mu_t,f_{ts})_{s,t\in T}$. The conditions ...
1
vote
0answers
95 views

Converse for Levy's continuity theorem

Levy's continuity theorem states that, for a sequence of random variables $\{X_n\}$ with characteristic functions $\{\varphi_n(t)\}$ and a random variable $X$ with a characteristic function ...
5
votes
1answer
114 views

Rate of convergence of Bayesian posterior

Suppose a data generating process (DGP) is parameterized by some unknown parameter $\theta_0$, say $P_{\theta_0}$, and we want to estimate the value of $\theta_0$ using Bayesian method. Let ...
5
votes
2answers
109 views

Finding joint probability from double marginals

Consider three probability distributions in the form $p_1(y,z),p_2(x,z),p_3(x,y)$. When does a global joint probability $p(x,y,z)$ (possibly not unique) exist? The first compatibility condition to ...
4
votes
0answers
77 views

Concentration of measure for uniform distribution on Stiefel manifolds

This is my first post on MO, so I hope the question is suitable. I am looking at the uniform distribution on the Stiefel manifold, but more specifically, at the uniform distribution on the ...
5
votes
1answer
192 views

Average probability that a random cosine polynomial with bernoulli coefficients is small

Let $P_{n}(t)=\sum_{k=0}^{n}\varepsilon_{k}\cos(kt)$ where $\varepsilon_{i}$ are independent random variables taking values in $\left\{-1,1\right\}$ with equal probability. Is is true that for any ...
4
votes
2answers
166 views

Probable direction of deviations from the expected value in binomial and hypergeometric cases

Suppose I have an urn with N marbles, with frequencies p and q for red and black marbles, and with p > 0,5. I take a sample of r marbles. It sounds intuitive to say that deviations from the mean ...
4
votes
0answers
65 views

Is there a generalization of Polya urns to continuous outcome event?

Take for example the simplest model where there are n blue balls and m white balls in an urn. Then, in a first step realization, a white one has been drawn and then c + 1 of this colour had been put ...
0
votes
0answers
74 views

absolutely continuous of two probability measures

Suppose $X_t$ satisfies $$X_t=\int_0^t b(X_s)ds+ L_t,\quad t\in[0,1]$$ where $L_t, t\in[0,1]$ is a $\alpha-$stable process. Let $P_L$ be the law of $L$, $P_X$ be the law of $X$. ($P_L, P_X$ are ...
1
vote
1answer
89 views

A differential inequality and a special value

Let $G \colon [0,1] \to [0,1]$ be a monotonically decreasing function with $G(0) = 1$ and $G(1) = 0$. Suppose that $G$ is differentiable infinitely many times, and that: $$G(x)G''(X) \leq ...
5
votes
1answer
118 views

Sufficient conditions for establishing a total order on a family of probability distributions?

Let $\mathcal{X}$ be some set of independent random variables. Define the ordering on $\mathcal{X}$ by $X_i \prec X_j$ if and only if $\mathcal{P}\left\{X_i \le X_j\right\} \ge \frac{1}{2}$. Are there ...
1
vote
1answer
189 views

Computing probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s

This question came up in my research: What is the probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s? So far I only figured out that I can do Monte ...
1
vote
2answers
99 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
1answer
107 views

Proof for additivity of cumulants

If one does not define cumulants via the cumulant generating function (cgf), e.g. because the cgf does not exist, then an alternative way is to use the recusion \begin{align*} ...
1
vote
0answers
56 views

Subclass of semimartingales for which all characteristics can be estimated?

I'm going to ask the question for Ito semimartingales rather than semimartingales in general, but more general answers would be great. An Ito semimartingale is a martingale for which the ...
2
votes
1answer
140 views

How to choose a random proper coloring

I am studying proper colorings of complete bipartite graphs and I'd like to be able to pick a random proper coloring and the compute some things about it. Recall that a proper coloring of a complete ...
5
votes
0answers
80 views

Real Zeros - tail estimate

Given a random polynomial with Gaussian coefficients, the Kac-Rice formula tells us what the expected number of real zeros is (for more on this, see the excellent paper of Edelman and Kostlan in the ...
11
votes
2answers
432 views

Maximum occupancy balls in bins with limited independence

Throw $n$ balls into $n$ bins and let $X_n$ be the maximum occupancy. That is the maximum number of balls found in any bin. If you throw the balls uniformly and independently it is known that ...
4
votes
2answers
118 views

Anti-concentration for sums of geometric random variables

Consider the random variable $Y = Y_1 + \dots + Y_k$, where each $Y_i$ is iid distributed as a geometric random variable with sucess probability $p$; here we should think of $p$ as being close to ...
1
vote
2answers
225 views

Variance of truncated normal distribution

Let $ X \sim \mathcal{N} ( \mu, \sigma^2 ) $, $ - \infty \leqslant a < b \leqslant +\infty $ ($ a, b \ne \infty $ simultaneously) and $ Y $ has a truncated normal distribution on $ (a, b )$, i.e. ...
1
vote
2answers
222 views

Existence of strong solution to SDEs with non-Lipschitzian drift

Consider the SDE: $$dX_t=b(X_t)dt+dW_t\quad X_0=x$$ If $b$ is bounded Borel function, using Zvonkin's Transform, one can prove there exists a unique strong solution. I want to know if we assume $b$ ...
3
votes
1answer
139 views

Determining the Fourier transform

Let $d>2$. Let $M$ be a 2-dimensional submanifold of $\mathbb{R}^d$. For instance (and this is the type of example I primarily care about) we could have $M$ being the set of scalar multiples of a ...
2
votes
0answers
42 views

Bounds on moving average process

Let $X_1,X_2,\dotsc$ be a sequence of i.i.d. random variables and define the average process $\{Y_t\}$ as $$ Y_t = \sum_{i=1}^p a_k X_{t-i} $$ with some constants $a_1,\cdots,a_p \in \mathbb{R}$. This ...
3
votes
1answer
65 views

Relatively compact sets in Ky Fan metric space

Let $(\Omega,P,\mathcal{F})$ be a probability space. $X$, $Y$ are two random variables. The Ky Fan metric defined as: $d_F(X,Y)=\inf\{\epsilon: P(|X-Y|> \epsilon)<\epsilon\}$ (or $d'_F(X,Y)=E ...
0
votes
0answers
88 views

Estimating the number of colors in a bucket

This question was previously posted to Math Stack Exchange here. Suppose we have a bucket containing a large (but known) number of balls. Each of the balls has a color. We don't know how many ...
4
votes
0answers
75 views

Is Wiener's Tauberian theorem true in Wiener space?

Let $\gamma$ be the standard product Gaussian measure in $\mathbb{R}^\infty$, and let $\mu$ be a finite variation measure, not necessarily positive, such that $\mu \ll \gamma$. Is the following true? ...
1
vote
0answers
100 views

Topological properties of space of Radon measures

Let $M$ denote the space of signed unbounded Radon measures on $\mathbb{R}$ as is defined by Bourbaki, i.e. $M$ is the dual of $C_c$ where $C_c$ is the space of continuous functions on $\mathbb{R}$ ...
2
votes
2answers
147 views

Is this a sufficient condition for joint normal distribution?

Suppose I have a random vector $\boldsymbol{Z}$, if I can prove that for $\forall \boldsymbol{\lambda} \neq \boldsymbol{0}$ where $\boldsymbol{\lambda}$ is a fixed vector, not a random vector, ...
13
votes
1answer
317 views

Smallest $k$ so that $k$-wise independence guarantees a constant expected minimum

Imagine you sample $n$ numbers with replacement uniformly from the integers $1,\dots, n$ (we can assume $n$ is large). Let $X$ be the minimum of these samples. I am interested in $\mathbb{E}(X)$ but ...
1
vote
0answers
31 views

Bound on the total variation distance for multiple samples $d_{tv}(P^n,Q^n)$

Given two discrete distributions $P$ and $Q$, with computable total variation distance $d_{TV}(P,Q)=||P - Q||_1$, is there a precise bound for $d_{TV}(P^n,Q^n)=||P^n - Q^n||_1$, as need to estimate ...
2
votes
1answer
75 views

Berry-Esseen bound in 2 dimensions for linear combinations

Let us say have a sequence of $n$ 2-$D$ random variables $X_i=(\varepsilon_i/\sqrt{n},i\varepsilon_{i}\sqrt{6}/n^{3/2})$, where $\varepsilon_{i}$ are independent random variables such that ...
0
votes
0answers
37 views

Tail inequality for orthomartingales/martingale difference random fields

It is known that if $(S_i= \sum_{j \leqslant i }X_i, \mathcal F_i)$ is a martingale, then for each $ \beta>1$, $\delta\in (0,\beta-1)$ and $\lambda>0$, and each integer $N \geqslant 1$, the ...
10
votes
2answers
306 views

Minimal expected absolute value of linear combinations of Gaussian random variables

I am interested in the following question. Consider $n$ independent standard normal random variables $g_i$. Cosider a linear combination $w_1g_1+\cdots+w_ng_n$. Can one give a "decent" upper bound for ...
0
votes
2answers
221 views

Generalized expression for balls and bins problem

$n$ number of balls are thrown randomly to $m$ number of bins, standing in a row. The balls are labeled as $1,2,3,....n$ and bins are also labeled as $1,2,3,...,m$. The probability of $i_{th}$ ball ...