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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44 views

How to check if a symmetric random variables is the difference of two iid symmetric random variables

I have the continuous symmetric random variable $X$ in $\mathbb{R}$. If I know its distribution function $F(x)$ what are the conditions on $F(x)$ so that $X=Y_1 - Y_2$ where $Y_i$ are also iid ...
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1answer
42 views

Gaussian expectation of a exponentiated outer product

Given a normal random column vector $\mathbf{x} \sim N(\mu, \Sigma)$, I need the expectation, $$ E\left[ \exp(\mathbf{xx}^\top)\right]$$ where $\exp(\cdot)$ is element-wise exponential function (not ...
-1
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0answers
32 views

On a sum statistically independent of its term [on hold]

Suppose $U$ and $V$ are two non-degenerate random variables, say real-valued for simplicity. Suppose further that their sum, $U+V$, and one term, $U$, are statistically independent. This happens when ...
2
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3answers
104 views

Estimating the Variance of a Discrete Normal Distribution

Let $f(x; \sigma) = \frac{1}{\sigma\sqrt{2\pi}}\cdot e^{-\frac{x^2}{2\sigma^2}}$ be the probability density function of a normal distribution $\mathcal{N}(0, \sigma^2)$. We consider a discrete normal ...
1
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1answer
55 views

Minimum of Random Energy Model (REM) with logarithmically correlated potential

In the paper [FB] (ArXiv, J. Phys. A), the authors analyse a particular Random Energy Model (REM) with logarithmically correlated potential and conjecture in Eq. (2) that the distribution function of ...
2
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0answers
54 views

Regularity of finite variation kernels in the (intersection) of the semimartingale spaces $H^p$

Suppose you have a continuous semimartingale $S_t=M_t + A_t$ where $A_t$ is the continuous finite variation part which has the form $A_t = \int_0^t b_s \, \mathrm{d} s$, where $\int_0^{\infty} |b_s| ...
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0answers
45 views

Expected value when rolling multiple k-sided dice and keeping the highest score and 1s cancelling higest remaining values [on hold]

sorry for the long title. I think the question is explained there, but I will go a bit further. I know how to calculate the expected value of n k-sided dice and keeping the highest score. If I am not ...
3
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0answers
41 views

Stochastic integration by parts to obtain Kailath Segall identity for iterated stochastic integrals?

If $(M_t)_{t \geq 0}$ is a continuous local martingale, one can define the iterated integrals $I_0=1$, $I_1(t)=M_t$ and for $n \geq 2$ $$I_{n}(t) = \int_0^t I_{n-1} (s) \mathrm{d} M_s.$$ By noting ...
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0answers
50 views

Convergence of empirical random variable

Let $X$ be a RV on the real line, of probability measure $P_X$, and let $X_n$ for $n=1,...,N$ be an iid sample from $P_X$. The Glivenko-Cantelli theorem says that the empirical measure, $P_N$, ...
3
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0answers
86 views

Birkhoff Ergodic Theorem and Ergodic Decomposition Theorem for Continuous-Time Markov Processes

I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to ...
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1answer
108 views

Steady state expectation of dynamic system of urns & balls

We have a large number of urns $N+1$. (Large means that the relative difference between $N$ and $N+1$ is well within the error bounds that I care about. The reason for the $+1$ will be apparent ...
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0answers
47 views

Quadrilaterals from a Unit Stick

This question could be seen as a coordinate-free variant of Sylvester's Four Point Problem (cf e.g. http://mathworld.wolfram.com/SylvestersFour-PointProblem.html): Suppose one are given an ...
1
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1answer
110 views

Stability of convergence in distribution under randomization

Suppose you have a sequence of non-negative stochastic processes $(X^n)_{t \in \mathbb{R}}$, $n \geq 1$, with continuous paths and continuous in $t$ such that $$\int_{-\infty}^{\infty} X^n_t \, ...
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0answers
41 views

Basic Probability Conditional Expectation [closed]

In my personal interests I read a introduction book to theory measure and probability with exercises. In order to understand definiton of conditional expectation I'm trying to solve those exercises ...
2
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1answer
128 views

Can't figure out “standard application” of the Garsia-Rodemich-Rumsey Lemma

I'm currently reading the paper http://arxiv.org/abs/0908.2473 and can't figure out what they call a "standard application" of the Garsia-Rodemich-Rumsey lemma (see p.8). Summed up, they have a ...
4
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0answers
110 views

Sum of a random number of identically distributed but dependent random variables?

Background Let $X_t$ be the continuous time Markov process on the state space {Working, Broken} with failure rate $\alpha$ and repair rate $\beta$. By elementary calculations [1] $$ \begin{align*} ...
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0answers
139 views

Measure concentration for law of large numbers

The classical law of large numbers states that $$\frac1k\sum_{i=1}^k X_i \rightarrow \mathbb{E} X_1$$ for i.i.d. $X_1, X_2, \ldots$ with finite $L^1$ norm. I was wondering whether is it possible to ...
0
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1answer
70 views

Probability of k overlapping subsets in N trials

Ok, here is what I am attempting to find an answer to: I draw M uniformly random subsets of size K from the set of numbers $\Omega=\{1, \dots, N\}$ (where uniformly random means that each unique ...
2
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3answers
103 views

Conformal invariance of Brownian motion in higher dimensions

We know for planar Brownian motion, that conformal maps composed with Brownian motion are also Brownian motion (preserve distribution). Does it follow for higher dimensions? I think it follows for ...
2
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0answers
46 views

Almost sure transversality of smooth random maps

I still am novice as far as probability is concerned and after fruitlessly Googling for an answer for a few days I thought I might have a better chance with MO. Let me first formulate the ...
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1answer
73 views

Residual lifetime of heavy-tailed random variable

The residual life time distribution of a random variable $X$ with distribution function $F$ is given by the formula \begin{equation}R(t)=P[X_\text{res}\leq t] = ...
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0answers
68 views

Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$

My question is motivated by an optimal transportation approach to PDE's and gradient flows in metric spaces (see e.g Otto's geometry of dissipative evolution equations: the porous media equation and ...
4
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0answers
99 views

Nontransitive dice

In the wikipedia article https://en.wikipedia.org/wiki/Nontransitive_dice it is claimed that " The set of nontransitive dice were investigated by the Latvian computer scientist and mathematician ...
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0answers
54 views

Dominating Poisson with parameter depending on a Bernoulli

Fix $\mu >0$ and take $\lambda \geq 0$. Let $B_p \sim \text{Ber}(p)$ with $p = \exp(-\mu - \frac{\lambda}2) $. Define the random variable $Y$ which is Poisson with parameter depending on the value ...
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3answers
192 views

Do regular conditional distributions almost surely assign trivial measure to all members of the conditioning $\sigma$-algebra?

Let $(X,\Sigma)$ be a standard measurable space, let $\rho$ be a probability measure on $(X,\Sigma)$, and let $\mathcal{E}$ be a sub-$\sigma$-algebra of $\Sigma$. We will say that a stochastic kernel ...
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0answers
38 views

Sets Closed under Stochastic Dominance Ordering

I'm working on a problem involving stochastic dominance and ``minimums'' of sets of random variables. For concreteness, consider two distributions with cdfs $F(x)$ and $G(x)$. We say that $F$ ...
2
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0answers
57 views

Reference request: Stochastic integration and martingale theory on the whole real line

I'm looking for a thorough treatment of stochastic integration and/or martingale theory on the whole real line, i.e. a way to construct a Brownian motion $(B_s)_{s \in \mathbb{R}}$ (if a two-sided BM ...
1
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1answer
95 views

Invertibility of random Vandermonde matrix

Let $\kappa, d \in\mathbb{N}$ and $f$ is a uniform probability measure on $\mathcal{D} = \left[-1,1\right]^{\kappa}$. In addition, let \begin{equation*} p = p\left(\kappa,d\right) := ...
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0answers
32 views

Tail Bounds for the minimum value of a function

Consider y to be the minimum value of an objective function over some subspace. More specifically $y= \min_x \|e+Bx\|_\infty \quad s.t. \quad x\in \mathcal{S}$ where $e$ is a known vector, $B$ is a ...
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1answer
130 views

Bounds on the moments of the binomial distribution

I'm looking for simple and reasonably tight bounds on the k-th moment of the Binomial distribution $B(n,p)$, namely, $E[B(n,p)^k]$. I'm interested in the case when k is large (say on the order of ...
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1answer
75 views

Measurable functions lifted onto a space of point measures are measurable

I've been reading [1] and attempting to prove statements given without proof. In the paper the authors construct a measurable space of measures over a base space, and as an aside show an elegant way ...
6
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1answer
638 views

Probability that a positive integer is the euler phi function of another positive integer

Define $f(n) = |\{m : m\le n, \exists k \text{ s.t. }\phi(k) = m\}|$. Clearly, $f(n)\le \left\lfloor \frac{n}{2}\right\rfloor + 1$ since $\phi(n)$ is even for all $n > 2$. Is ...
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1answer
62 views

softening probability distribution function

I am working on ECG signals and I want to fit it's probability distribution function with gaussian mixture model (sum of 2 or 3 gaussians) to extract features but it has a very sharp pdf around zero. ...
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1answer
176 views

Geometry description of the GSR riffle shuffle model

In 1992 Diaconis and Bayer announced their famous result which is now a well-known folklore: Seven shuffles is enough to randomize a deck of cards. One of the key ingredients in their proof is that ...
3
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1answer
99 views

What is the probability of a given induced ordering of a random permutation?

I ran into the following problem in a calculation involving permutations. Let $[n] = \{1,...,n\}$, and assume that $[n]$ is partitioned into equivalency classes. That is, $[n]$ is the disjoint union ...
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0answers
40 views

Reference for “Newtonian capacity estimates probability that A is hit by a Brownian motion”

I am looking for the following statement "In fact, the Newtonian (logarithmic) capacity gives an estimate, up to a constant factor, the probability that A is hit by a Brownian motion started, say, ...
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1answer
286 views

Basketball shots and stopping rule

Moved over from StackExchange. You are taken to play a basketball game where you can shoot basketballs at n slots using a machine that is equally likely to shoot the balls into those n slots. You can ...
1
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1answer
177 views

Double Markovity

Suppose we have a double Markov relation for three random variables $X$, $Y$ and $W$ as follows $$X\to W\to Y,$$ and $$X\to Y\to W.$$ How to prove that there exist functions $f$ and $g$ such that ...
4
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0answers
85 views

First return time in an interval for N particles rotating on the circle at constant random speeds

Here is my problem: draw N velocities $v_1,v_2,\dots,v_n$ in $[-\pi,\pi]^N$ from some measure (Haar measure of uniform independent for simplicity) and make $N$ particles rotate around the circle with ...
3
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1answer
180 views

An inequality concerning convexity and expectation

Assume $f$ and $g$ are nonnegative with $$\int_0^\infty f(x)dx=1=\int_0^\infty g(x)dx $$ and $$\int_0^\infty xf(x)dx<\infty > \int_0^\infty xg(x)dx $$ Is it true for nonnegative numbers $p$, $q$ ...
5
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2answers
141 views

Pairwise dependent random walk recurrent

Let $\{D_i\}_{i=0,1,2,\dots }$ be independent $\exp(1)$ random variables. We use the collection $\{D_i\}$ to define a random walk on $\mathbb Z$ by $S_0 = 0$ and $S_n = \sum_1^n X_i$ with $X_i \in ...
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1answer
69 views

Mutual information staying constant under composition of channels

Consider the following scenario: one has 2 communication channels $C_1$ and $C_2$. Denote by $p(x)$ the input probability distribution. The mutual information between the input and the output of ...
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0answers
407 views

The lonely molecule

Suppose $n$ air molecules (infinitesimal points) are bouncing around in a unit $d$-dimensional cube, with perfectly elastic wall collisions. Let $k=n^{\frac{1}{d}}$. For example, in 3D, $d=3$, with ...
5
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1answer
110 views

non-negative random variable

Let $X$ be a real-valued random variable with $X \geq 0$ and $\mathbb E X >0$. I would like to bound $\mathbb P(X >0)$ from below using information about the first few moments of the variable. ...
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3answers
580 views

A conjecture about the entropy of matrix vector products

Consider a random $n$ by $n$ circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$. Let $M'$ be the $m$ by $n$ matrix which is formed by taking the first $m$ rows of ...
2
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1answer
163 views

Does the set of automorphisms of a cyclic group exhibit some sense of randomness?

I prefer to proceed with a concrete example if I may. I appreciate that the answer might well be better explained with group theory, geometry and/or notions from probability theory, which I welcome. ...
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3answers
929 views

Not-lonely runners

The lonely runner conjecture has several formulations. They all involve a number $n$ runners running on a circular track, each with a different speeds, and the conjecture is that each runner is ...
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2answers
83 views

Is this generating family of a measurable space of point measures a pi-system?

I'm learning some probability and measure theory and working my way through the first few paragraphs of [1]. My question is perhaps too basic for Math Overflow, but I hope it is welcome here. Point ...
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13 views

Set of expected values [migrated]

I have a doubt, I'd like to calculate a the expected value of a set, let suppose I have a set of n points, every point has a probability p_i, the expected value of this set is the sum of all p_i ...
2
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1answer
73 views

limiting empirical spectral distribution of the Laplacian matrix on an Erdos-Renyi graph?

Let $G$ be an Erdos-Renyi random graph (i.e. an edge ($ij$) exists with probability $0 < p < 1$ and all edges are independent). Let $L$ be the Laplacian matrix of this graph (i.e $L=D-A$, where ...