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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4
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1answer
175 views

An intutive reason why a “distance” metric may be a poor one for a procedure where we attempt to modify a string (mutating 0 OR 1 bits)

If I'm attempting to mutate one arbitrarily chosen binary string $s_a$, to another arbitrarily chosen binary string $s_b$, in the smallest number of steps (i.e. with the smallest number of mutations) ...
1
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0answers
191 views

Question on measure zero set of initial conditions in dynamical systems

[Update] Let $S \subseteq \mathbb{R}^n$ be a closed, bounded, convex set with measure $m(S)>0$ and let an autonomous dynamical system (system of ODEs) be given by $$\frac{dx}{dt} = f(x),$$ where ...
14
votes
2answers
366 views

How to sample uniformly from singular matrices

I would like to uniformly sample from all singular $n$ by $n$ Bernoulli matrices (that is each entry is $1$ or $0$ with probability $1/2$). I could of course just sample from all $n$ by $n$ Bernoulli ...
2
votes
1answer
61 views

Relating joint probability to norm of vector of probabilities

I have a set of Bernoulli random variables $X_1,X_2,\ldots, X_n$ and I would want to bound the joint probability $P(X_1=0,X_2=0,\ldots, X_n=0)$ using the norm $\lvert \lvert \mathbf{p}\rvert \rvert$, ...
1
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1answer
122 views

a question about the proof of identification of dual space

I have a question about the proof below: Let $\mathcal{C}:=\mathcal{C}(\mathbb{R})$ be the space of continuous functions on $\mathbb{R}$ and $\mathcal{C}_b$ its subspace consisting of bounded ...
8
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3answers
352 views

References request: constructive quantum field theory

I am taking a course this semester on QFT, which deals much with constructive quantum field theory. Some of its topics so far involve relationships between non-Gaussian probability measures,Feynman ...
6
votes
1answer
175 views

The time to drift a binary string from one state to another via deterministic selection of two possible random bit mutation procedures

I have some length $L$ binary string consisting of an ordered array of bits: $(b_1, b_2, ..., b_{L})$, where all bit values $b_i$ are originally set to zero. I'd like a particular set of $n$ bits to ...
5
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0answers
112 views

Semi-directed first-passage percolation on $\mathbb{Z}^2$ with deterministic vertical weights

Consider the following first-passage percolation problem on the plane grid $\mathbb{Z}^2$: all the horizontal edges are directed (pointing east) and carry an independent random weight, say standard ...
2
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1answer
118 views

Transition probabilities in coupled Markov chains

I know that for a continuous-time Markov chain, the probability of transition from time $0$ to $t$ is given by $P(t)=e^{Q(t)t}$. I have a system of $N$ interdependent continuous-time Markov chains ...
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0answers
67 views

equivalence of topologies defined on $M_1$(a subspace of bounded measures on $\mathbb{R}$)

Let $\mathcal{C}:=\mathcal{C}(\mathbb{R})$ be the space of continuous functions on $\mathbb{R}$ and $\mathcal{C}_b$ its subspace consisting of bounded elements. Define for $\phi(x):=1+|x|$, $$ ...
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0answers
440 views

Probability a random Toeplitz matrix is singular

Consider Toeplitz matrices where the entries in the first row and column (which define the whole matrix) are independently chosen to be either $1$ or $0$ with probability $1/2$. Define $p_n$ to be the ...
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1answer
51 views

Minimal variance for phase-type distributions?

Let $\mathcal{D}(m)$ be the set of phase-type distributions constructed from $m+1$-state Markov chains. Recall that the coefficient of variation of a distribution $D$ is the ratio of the standard ...
4
votes
1answer
117 views

Epidemic threshold

Need some help / ideas to proceed. Stuck for a while on this. In the literature of epidemic theory, it is found that the epidemic threshold is $1/\lambda_{max}(A)$ where $\lambda_{max}(A)$ is the ...
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0answers
57 views

Berry-Esseen result for triangular arrays

Let $\{\{X_{n1},\ldots,X_{nn}\},n=1,2,\ldots\}$ be a triangular array of independent random variables where each row contains identically distributed random variables. Let $E[X_{n1}]=\mu_n<\infty$ ...
5
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0answers
123 views

Distribution of subset-sums

Let $A$ be a set of $n$ integers uniformly distributed in $\{0,\dots,N-1\}$. Let $S$ be the set of subset-sums modulo $N$ of $A$. Let $f_{n,N}(k)$ be the probability that $|S|=k$. Is there an ...
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0answers
191 views

What is the characteristic functional for Brownian motion on a sphere?

I'm a physicist, somewhat familiar with stochastic processes, but I'm a little unsure of what follows. What I basically have is a complicated quantity involving a vector that is equivalent to ...
3
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0answers
148 views

Maximization of a total variation distance subject to another total variation distance in Markov chain

Suppose two dependent random variables $X$ and $V$ from finite alphabets $\mathcal{V}$ and $\mathcal{X}$ with known joint and marginal distributions are given. Let $P_{XV}$ and $P_X$ and $P_V$ are the ...
4
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0answers
149 views

Maximizing Renyi entropy for a certain channel

The channel under consideration is $T = A + B$, where $A$ and $B$ take on values in $\{0, 1\}$ according to a probability mass function. Let (joint) random vector $(A_1, A_2,\ldots, A_n)$ be denoted ...
3
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0answers
40 views

Relative vulnerabilities in SIS epidemic model

Consider the SIS model of epidemic spreading. There is a finite graph $G(V,E)$, link infection rates $\lambda_{ij}$ and node recovery rates $\mu_i$. There are a few initial nodes which are infected at ...
4
votes
1answer
134 views

Log-likelihood in regime switching

Not sure if this question is too simple to be asked here... In the following paper Cho, Jin Seo, and Halbert White. "Testing for regime switching." Econometrica 75.6 (2007): 1671-1720. doi: ...
4
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1answer
192 views

Population dynamics for fish arriving via a Poisson process and living for a time given by some (not necessarily symmetric) general distribution

Imagine we have a hypothetical population of fish in a pond. The fish cannot reproduce, but are introduced by a Poisson process (with some known and fixed rate parameter independent of the total ...
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1answer
66 views

Existence of a map connecting two marginals of a product measure

Let $X$ and $\bar X$ be two standard Borel spaces, and let $A\subseteq X\times\bar X$ be an analytic subset of the product space. Let $P$ be any probability measure such that $P(A) = 1$, and denote by ...
7
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2answers
515 views

Probability two matching runs of coin tosses

If you toss a coin $2\ell-1$ times you get a sequence of outcomes, say, $HTHTHTH$ for $\ell = 4$. I am trying to work out the probability that there are at least two runs (in other words contiguous ...
2
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0answers
62 views

Local limit theorem for the exponential of chi-squared random variables

Let's denote a chi-squared distribution with $k$ degrees of freedom as $\chi^2_k$, and a random variable $Z_k=\frac{X_k-k}{\sqrt{2k}}$, where $X_k\sim\chi^2_k$. Thus, $Z_k$ is a chi-squared random ...
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0answers
49 views

Asymptotic symmetry of distributions

Suppose we have a triangular array of positive random variables $\{\{X_{1i},\ldots,X_{ii}\},i=1,\ldots,n\}$ such that the random variables in the array are independent, all random variables have mean ...
3
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0answers
94 views

Quantile convergence

Let $X^1,\dots,X^n$ be a sample of (not necessarily iid) random variables and denote $$F^n(x)=\frac{1}{n}\sum_{i=1}^n \mathbf 1_{X^i\leq x}$$ the empirical distribution function. Suppose that we know ...
7
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0answers
90 views

Neigbourhood of Poisson point process

Let $C$ be a unit circle and $P$ be a Poisson point process on $C$ with intensity $1$. For each point $p$ of the process we draw an interval $(p-\epsilon, p+\epsilon)$ for some $\epsilon>0$. ...
0
votes
1answer
116 views

What is the area of the piece of an $n$-sphere within a given angle of a vector? [closed]

Let $x$ be the unit vector $(1,0,0,\ldots,0)$ in $\mathbb{R}^n$, and let $A(\theta)$ be the subset of $\mathcal{S}^{n-1}$ whose angle to $x$ is less than $\theta$, i.e. $$ A(\theta) = \left\{ y \in ...
2
votes
1answer
133 views

A question about stochastic kernels and invariant measures

Suppose that $E$ is a metric space, let $\mathcal{B}_E$ denote the set of its Borel subsets and suppose that $\mu$ is a probability measure on $(E,\mathcal{B}_E)$. In addition, suppose that $p:E\times ...
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0answers
76 views

An extension of first order stochastical dominance property

A random variable $X$ (distributed as $F$) is said to be stochastically larger than $Y$ (distributed as $G$), ($X>_{FSD}Y$) if their distribution functions satisfy $G(y)>F(y)$ for all $y$. It ...
2
votes
1answer
84 views

Question about infinite-dimensional BM

Suppose we are given an $L^2(\mathcal{D})$-valued Brownian motion $W_t$ defined by $$W_t:=\sum_{k=1}^{\infty}\sqrt{\sigma_k}W_t^k\phi_k(x),$$ where $\mathcal{D}$ is bounded domain in $\mathbb{R}^d$, ...
4
votes
1answer
88 views

Weak ergodicity of nonhomogenous products of 0-1 matrices

Here is a question which probably has a negative answer, but I couldn't find any literature directly on it. Let $(A_n)$ be a sequence of rectangular 0-1 matrices (that is, the entries are restricted ...
5
votes
1answer
95 views

Deviation bound for the maximum of the norm of Wiener process

Let $W(t)$ be an $n$-dimensional Wiener process. Denote by $\chi_n^2$ a chi-squared random variable with $n$ degrees of freedom. I have recently found the following inequality given without proof: $$ ...
2
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1answer
78 views

Conditions for existence of $m$-th differentiable root of a non-negative definite matrix

In M.I. Friedlin's famous paper "On the Factorization of Non-Negative Definite Matrices", he shows that if a non-negative definite symmetric matrix $a(x)=\{a^{ij}(x)\}_{i,j=1}^n$ is in ...
5
votes
1answer
171 views

“Smallest” event such that probability greater than a given value

Very briefly, consider the probability space $(\mathbb R^n, \mathcal{B}(\mathbb R^n),P)$. During a problem I am studying, I came to a point where i need to compute \begin{equation*} \begin{aligned} ...
2
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0answers
80 views

On the infinitesimal generator of a 1-dimensional stochastic heat equation: core and explicit form

Denote $E = C([0, 1])$. I am consider a 1-dimentional stochastic heat equation on $h$: $\partial_tu(t, x) = \partial_x^2u(t, x) - V'(u(t, x)) + \dot{W}(t, x)$, for all $(t, x) \in (0, ...
2
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2answers
163 views

Probability of random geodesics on the round sphere intersecting

4 end points (a,b,c,d say) are chosen uniformly randomly and connected a to b and c to d by two geodesics on the 2-dim round sphere. Here uniformly is in the obvious sense of the volume form ...
4
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1answer
148 views

A queuing process where customers must be detected

Imagine a scenario where customers arrive in some queue according to a Poisson process with rate parameter $\lambda_{arr}$, and where the process of responding to the customers has a kind of ...
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2answers
336 views

Quantization by cohomology

Ok, so I have heard some cool stuff here and there about how to Quantize Yang-Mills via cohomology, can anyone refer any texts in the literature that have shed some light on this, I mean I have some ...
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0answers
122 views

An optimization in Markov Chain

We are given two correlated random variables $V$ and $X$ supported over a finite alphabets $\mathcal{V}$ and $\mathcal{X}$. Suppose the marginal $P_V$ and conditional distribution $P_{X|V}$ are ...
3
votes
1answer
179 views

The pdf of of the determinant of an interesting random matrix

Consider a matrix of order $n$ whose elements are independently and uniformly chosen from the interval $[a,b]$ wher both $a$ and $b$ are positive .Let $X$ be the r.v. corresponding to the determinant ...
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0answers
58 views

Discreet customers at more or less discrete counters

A bank has $N$ counters in a row, and customers arrive irregularly at an average of 1 per minute (say, according to a normal distribution with variance $\sigma^2$ – but I don't think that is ...
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1answer
186 views

Stationary distribution in general Markov Chains

This is just a reference request for a result which is very general, useful and should be well-known, but I've failed to find a good reference to cite. The problem is to define the "most natural" ...
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1answer
321 views

Prove or disprove $ \int_{0}^{\infty} \int_{-x}^{0} f(x)f(y)dydx > \int_{0}^{\infty} \int_{-\infty}^{-x} f(x)f(y)dydx. $

Consider a symmetric, unimodal distribution $f(x)$ such that $\int_{0}^{\infty} f(x) > 1/2$. I want to prove or disprove the following: $$ \int_{0}^{\infty} \int_{-x}^{0} f(x)f(y)dydx > ...
0
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1answer
105 views

On the expected value of a random integral:

Is it possible to find the expected value of $u(t)$ in terms of the following information: $$u(t)=\int_{0}^{t}(t-s)(f(s)+(T-s)Y)X_sds$$ where: $X_s$ is a wide sense stationary process with known ...
2
votes
1answer
118 views

Computation complexity of calculating the cdf of an n-th dimensional gaussian random vector

Suppose you have a general $n$-th dimensional random Gaussian vector with probability distribution function $\mathcal{N}\left(\mathbf{x}|\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$. What is the ...
0
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1answer
183 views

Convergence of the empirical distribution function

Let $\alpha\in\mathbb R^d$, with $\alpha\neq 0$. Take a sequence of iid random variables $X^{1},\dots,X^{n}$ with values in $\mathbb R^d$ and denote $R$ the cdf of $\alpha X^1$ (where the product is a ...
3
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1answer
351 views

Simple reason that a mathematician cannot do better than random when guessing contents of a box?

I have a question about the finite analog of the puzzle proposed here involving mathematicians guessing the contents of boxes. Specifically, suppose there are $k$ unopened boxes each containing a ...
2
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1answer
229 views

Expected value of $(1 - X)^{-2} $ over Haar measure of the unitary group, $X \in U(N)$

Let $\lambda_1, \dots, \lambda_n$ be the eigenvalues of a random Unitary matrix. I am interested in the expected value: $$\mathbb{E}_{X \in U(N)}\left[ \prod_{i=1}^n \frac{1}{(1 - ...
0
votes
1answer
113 views

Expected rank of players in a Bradley-Terry round-robin tournament

Let $[n]$=$\{1,\dots,n\}$ be a set of players in a round-robin tournament. Each player $i$ has an associated skill parameter, $\lambda_{i}$, and the probability that player $i$ defeats player $j$ is ...