**4**

votes

**1**answer

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**

vote

**0**answers

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

**2**answers

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

**1**answer

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**

vote

**1**answer

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**

votes

**3**answers

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

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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 $\phi(x):=1+|x|$,
$$
...

**11**

votes

**0**answers

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 ...

**1**

vote

**1**answer

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

**1**answer

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 ...

**1**

vote

**0**answers

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**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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

**1**answer

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**

votes

**1**answer

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 ...

**0**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**0**answers

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 ...

**2**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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

**1**answer

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

**1**answer

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 ...

**0**

votes

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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**

votes

**1**answer

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

**1**answer

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**

votes

**0**answers

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**

votes

**2**answers

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**

votes

**1**answer

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 ...

**1**

vote

**2**answers

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 ...

**1**

vote

**0**answers

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

**1**answer

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 ...

**1**

vote

**0**answers

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 ...

**0**

votes

**1**answer

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" ...

**1**

vote

**1**answer

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**

votes

**1**answer

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

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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

**1**answer

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 ...