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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15
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3answers
380 views

An inequality for two independent identically distributed random vectors in a normed space

Suppose that $X$ and $Y$ are independent identically distributed random vectors in a separable Banach space $B$. Does it always follow that $E\|X-Y\|\le E\|X+Y\|$? Some background information on ...
1
vote
1answer
170 views

Is regularity closed under products?

Let $G \colon [0,1] \to [0,1]$ be a differentiable cumulative distribution function (monotonically non-decreasing function with $G(0) = 0$ and $G(1) = 1$). We say that $G$ is regular if $$ x - ...
11
votes
0answers
284 views

Probability of many overlapping zero inner products on a circle

[Question edited and changed a little on June 14 2015] Consider an $n$-dimensional vector $v$ with $v_i \in \{-1,1\}$. Now consider an $n$-dimensional vector $w$ with $w_i \in \{-1,0,1\}$. The ...
3
votes
1answer
87 views

Concentration and Correlation for Magnitudes of Gaussian Vectors

Suppose we have a large collection of standard normal random variables $a_i\in\mathbb{R}^n$. We know by standard concentration results that if we take $m \geq C\left(t/\epsilon\right)^2n$ samples, ...
2
votes
2answers
276 views

Suggestions for dealing with the “timed” balls-into-bins model

Definitions: Let $T$ (for "time") be a random variable $T \sim \text{Exp}(\lambda)$ and $\Delta t$ is a realization (or called an observed value) of $T$. Let $D$ (for "delay") be a random variable $D ...
1
vote
0answers
42 views

Bounds on the spherical measure of sub-level sets of quadratic forms

I'm wondering if there are any bounds on the spherical measure of sets of the form $$ \mu_n\left(\{y\in S^{n-1} : \frac{y_1^2}{y_2^2} < \alpha\}\right) \leq f(\alpha) $$ where $\alpha$ is some ...
4
votes
1answer
153 views

Why the term “geometric” rough path?

A "geometric" rough path is a rough path such that $Sym(\mathbb{X}_{s,t})=\frac{1}{2}X_{s,t}\otimes X_{s,t}$. For example the Ito rough path is not geometric because ...
1
vote
1answer
145 views

Arc Sine law for Random Walk conditioned to non-absorption or not?

Let $S_n$ be simple symmetric Random walk on the integers in $[-N,N]$ with states $N$ and $-N$ absorbing. Let $\tau$ be the time to absorption when $S_0 = 0$. Is the $E(S^{2}_{n}| \tau \geq n)$ ...
7
votes
1answer
180 views

Decompose dependent random variables into function of dependent and independent parts

Let $X$ and $Z$ be two (possibly dependent) random variables. Is it necessarily the case that there exists a Borel function f and a random variable $Y$ that is independent of $X$ such that $Z = f(X, ...
4
votes
0answers
177 views

What is the probability that a Gaussian random variable is the largest of three Gaussian random variables?

Consider three independent, normally distributed RVs: $YA \sim N(a,\sigma ^{2}),$ $% YB\sim N(b,\sigma ^{2})$ and $YC\sim N(c,\sigma ^{2})$. What is the probability that $YA$ is the maximum?: $$\Pr ...
4
votes
2answers
634 views

Polish by compact is Polish?

Let $X,Y$ be separable and metrizable, with $Y$ Polish, and suppose there is a topological quotient map $f:X\to Y$ with compact fibers. Is $X$ Polish? I have a specific space in mind, so if the ...
7
votes
1answer
262 views

understanding of rough path

A rough path is defined as an ordered pair $ (X, \mathbb X)$, where $X$ is a path mapping from $[0,T]$ to some Banach space $V$ and $\mathbb X:[0,T]^2 \mapsto V^2$ is another mapping for additional ...
10
votes
3answers
456 views

On the number of consecutive divisors of an integer

Define for $n \in \mathbb{N}$ the function $$\tau_1(n):=\sum_{\substack{d|n, \\ d+1|n}}1,$$ i.e. the number of consecutive divisors of an integer. The average of $\tau_1(n)$ is $1$ since $$\sum_{n\leq ...
0
votes
0answers
26 views

Truncated Robbins-Monro

I'm reading Han-Fu Chen's book "Stochastic Approximation and Its Applications", and in Chapter 1, he's got a statement of a theorem and proof on a truncated Robbins-Monro algorithm. In this version, ...
4
votes
1answer
91 views

Weakly correlated Bernoulli field

Let $\Lambda\subset\mathbb{Z}^{d}$ ($\Lambda$ is finite). Let $\left\{ \eta_{x}\right\} _{x\in\Lambda}$ be a field of dependent Bernoulli random variables. I assume that their correlation decays ...
1
vote
0answers
79 views

Scaling of First-passage times for Random Walk on integer lattices

Consider simple symmetric random walk $S_{n} = (S_{n}^{(1)},\dots, S_{n}^{(d)})$ on the d-dimensional integer lattice with starting point the origin. Let $\tau_{N}$ be the first time $S_{n}$ exits ...
18
votes
0answers
396 views

Two conjectures about zero inner products and dissociated sets

The following problems come from something I worked on (with my coauthors) related to proving a new time lower bound for streaming problems. Having worked on these problems for some time with little ...
0
votes
1answer
52 views

How can two random variables are continuous infers that their jointly random variable is continuous [closed]

We assume that $\forall a,b$ suchthat $a^2+b^2>0$, $aX+bY$ is continuous random variable. But we don't assume that $X$ and $Y$ are independent. My question is the following: Is it true that the ...
2
votes
0answers
64 views

Cramer-Rao type bound for absolute estimation error

Let $\{X_1, X_2, \ldots, X_n\}$ be independent and identically distributed (i.i.d.) random variables sampled from a common distribution with density $f_{\theta}(x)$, where $\theta$ is an unknown ...
2
votes
1answer
152 views

Random walk on a sphere along latitude-longitude grid

Suppose a sphere is partitioned by a latitude-longitude grid, with grid quadrilaterals $\Delta \times \Delta$. All grid nodes have degree $4$, while the North & South poles have degree $2 \pi / ...
11
votes
1answer
229 views

No limit shape for random Young diagrams under z-measure?

In their paper Random partitions and the Gamma kernel (Advances in Mathematics 194 (2005) 141–202), Borodin and Olshanski state that: An important difference between the Plancherel measures and ...
4
votes
1answer
160 views

Upper bound of the waiting time of a sum process

Let $n \in \mathbb{N}$, $x_1, \ldots, x_n \in (0,1)$ fix but arbitrary, s.t. $\sum_{i=1}^n x_i = 1$. Let $X_i \sim \operatorname{Unif}(\{x_1, \ldots, x_n\})$ i.i.d., and $T_n = \min\{t \in \mathbb{N} ...
4
votes
1answer
131 views

variance of compound binomial distributions

The below is motivated by a problem I'm observing in my experimental data I have m boxes, where each box is supposed to contain k molecules of mRNA. The measurement process includes labeling all the ...
4
votes
1answer
247 views

Strings with no long runs from proper subalphabets

Let $R_{n,k,b}$ be the number of $b$-ary strings of length $n$ that contain some run of length at least $k$ from some $(b-1)$-ary subalphabet. Let $N_{n,k,b}=b^n-R_{n,k,b}$ be the size of the ...
3
votes
0answers
73 views

Explicit formula for the moment problem with Carleman's conditions

Suppose that $\mu_j$ are real numbers obeying the Carleman condition: $\sum_{j=0}^\infty 1/\mu_{2j}^{2j}<\infty$. Then it is well-known that in case the $\mu_j$ are positive definite, there is a ...
0
votes
0answers
59 views

What is the concentration of measure for Gaussian random variables which are independent, but are transformed?

This might be a too easy question for Mathoverflow, but Googling led to similar questions and answers here (though not the one I was looking for). The question is split into two: I have a matrix $X ...
4
votes
2answers
289 views

“Typical” convergence rate for the von Neumann mean ergodic theorem

The von-Neumann theorem states that for a unitary operator $U: {\cal H} \mapsto {\cal H}$, where ${\cal H}$ is a Hilbert space, the following holds: $$ \lim_{N\to \infty} \frac{1}{N} \sum_{n=1}^N U^n ...
4
votes
0answers
112 views

Young Tableau Box Correlations

Let $T$ be a uniformly random Standard Young Tableau (SYT) of shape $\lambda=(\lambda_1,\cdots,\lambda_k)$ with $|\lambda|=n$. Let $T_{ij}$ denote the value in box $(i,j)$. I'm interested in what can ...
2
votes
1answer
123 views

What is the distribution of the maximum nearest-neighbor distance of a point cloud sampled from a solid body like?

Let $\mathcal{B} \subseteq \mathbb{R}^n$ be an $n$-dimensional solid body. Assume that we sample $N$ points, say $S = \{ x_1, ..., x_N \}$, from $\mathcal{B}$ uniformly at random. Consider the ...
5
votes
1answer
340 views

How many random matrices does it take to generate a matrix algebra?

Let $\mathbb{F}$ be a finite field. Let $A\le \mbox{Mat}_n(\mathbb{F})$ be a matrix algebra. Is there a good bound on the number $k$ of random elements $a_1,\dots,a_k\in A$ that one needs to take ...
18
votes
0answers
447 views

Human Knot game

In the popular game "Human Knot", a group of people stands in a circle and each person grabs another person's hands at random (one with the left hand and one with the right hand). The goal is to ...
31
votes
2answers
931 views

Random sequence of integers in $\{1, 2, \dots, n \}$ which is “everywhere probably increasing” - how long can it be?

Let $D=(d_1,d_2,\dots,d_k)$ be a sequence of correlated random variables. $D$ is "everywhere $r$-probably increasing" if the event $d_j > d_i$ has probability $\geq r$ for all $j > i$. Fix $r ...
4
votes
2answers
233 views

Is $B(t-1)$ an Ito process?

Let $I(\cdot)$ be an indicator, and $B_{t}$ be an 1-dim standard Brownian motion in a nice filtered probability space $(\Omega, \mathcal{F}, P, \mathcal{F}_{t})$. We consider a random process $$Y_{t} ...
1
vote
1answer
98 views

Probability of sub-sequence of exact length to occur

Let's suppose that I have a sequence of length $L$ of uniformly distributed random numbers on interval $(a,b)$. How can I calculate probability that increasing sub-sequence of length $M,M <L, $ ...
0
votes
1answer
103 views

probability in galton watson processes [closed]

I am trying to study the Elementary new proofs of classical limit theorems for Galton Watson processes written by Jochen Geiger. I don't understand what Z_(n,i) stand for. And in the proof of Theorem ...
-1
votes
1answer
109 views

Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$ [closed]

Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology. Let $X$ be a topological space (for convenience, it might be Polish ...
7
votes
1answer
216 views

Central limit theorem for biased random walk

Define random variables $X_n$ by $X_0 = 0$ and \begin{equation*}X_n - X_{n-1} = \begin{cases} 1 & \text{with probability } g(X_{n-1}) \\ 0 & \text{with probability } 1-g(X_{n-1}) \end{cases} ...
1
vote
0answers
120 views

Joint law of a standard Brownian motion and its local time at a nonzero level

Let $B_t$ be the standard Brownian motion and $L_t^a$ be the local time at level $a$. It is known that the joint-density of $(L_t^0,B_t)$ is $$ P\left(B_t\in d y, L_t^0\in d v\right) = ...
3
votes
1answer
594 views

Is the integral always nonzero?

Let $$I_{n,p,a}:=\int_0^{\infty-} \frac{g_{n,a}(t)}{t^{p+1}} \, dt,$$ where $$(*)\qquad\qquad\qquad n\in\mathbb N,\quad -\infty<a<\infty,\quad p_{n,a} < p < ...
0
votes
0answers
41 views

Recursive parameter estimation for partially observed Ito SDEs

I'm trying to get my head around online (recursive) maximum-likelihood parameter estimation in the language of stochastic processes and in the context of stochastic filtering, i.e. where we have a ...
2
votes
0answers
64 views

Poincare inequality for the measure of Brownian path

I am wondering if the Poincare inequality holds for the Brownian path space. As the simplest example, let $\{w_t, t \in [0, 1]\}$ be a 1-d standard BM: has independent increments and continuous ...
1
vote
0answers
96 views

Horizontal vs Vertical sides Exit from a Rectangle for simple symmetric Random Walk on $\textbf{Z}^{2}$

Consider simple symmetric random walk, $X_{n} = (X_{n}^{(1)}, X_{n}^{(2)})$ with $X_0= (0,0)$, on the 2 dimensional integer lattice, $\textbf{Z}^{2}$. Let $T_{M}, T_{N}$ be the smallest $n$ such ...
2
votes
0answers
41 views

Functions whose Laplace transforms have prescribed behavior at minus infinity

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a non-negative function with entire Laplace transform $\hat{f}$ (in particular $\lim_{t\to \infty}e^{st}f(t)=0$ for all $s$), and $p_0$ a positive integer. ...
4
votes
1answer
143 views

convergence of integral for each bounded function in probability

Let $\mu, \mu_1, \mu_2, \dots$ be random measures on a Polish space (separable completely metrizable topological space) $(S, {\mathcal S})$. Suppose I know that $$\int f d \mu_n \to \int f d\mu$$ ...
0
votes
1answer
207 views

Are such averages known with representations of $S_n$?

Like is there a sense in which one can quantify that for two group elements (in different conjugacy classes) their characters are "close" for some fixed irreducible representation? (feel free to ...
1
vote
1answer
141 views

Interchanging limits for doubly indexed random sequences

I've encountered the following problem which seems to be quite standard but for which I can't find any proper references (asking on mathematics SE didn't bring up any answers so I'm reposting my ...
8
votes
7answers
571 views

Finite-space dynamical systems

This question is quite open-ended, but I will formulate several sub-questions that I'll try to make precise. It is about finite-state dynamical system: start with a finite set $X$, with say $n$ ...
3
votes
1answer
75 views

random odes adapted solution

Let $\{\omega_t\}$ be a Levy process (like Brownian Motion, stable process). Consider the following random ode $$x_t=x_0+\int_0^tb(x_s+\omega_s)ds$$ Where $b$ is a bounded continuous function (not ...
13
votes
4answers
773 views

Random Diophantine polynomials: Percent solvable?

Suppose one generates a random polynomial of degree $d$ with integer coefficients uniformly distributed within $[-c_\max,c_\max]$. For example, for $d=8$, $|c_\max|=100$, here is one random ...
3
votes
0answers
223 views

Unusual generalization of the law of large numbers

I have seen in physical literature an example of application of a very unusual form of the law of large numbers. I would like to understand how legitimate is the use of it, whether there are ...