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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10
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1answer
433 views

Metric $d(A,B) = \mathbb P(\overline A\cup\overline B\mid A\cup B)$

I'm wondering where the relative probabilistic distance was first studied: $$d(A,B) =\mathbb P(\overline A\cup\overline B\mid A\cup B)$$ where $\overline A$ is the complement of $A$. A web search ...
1
vote
0answers
147 views

Threshold for perfect Matchings in Bipartite graph

Consider the random bipartite graph with vertex classes of size $n$ and each edge being present independently with probability $p(n)$. I know one way to prove the threshold of a perfect matching is ...
2
votes
2answers
105 views

PDF of the product of normal and Cauchy distributions

I am having trouble in finding out the resulting PDF of the product of normal and Cauchy distributions. It turns out that we have a general formula for calculating the PDF of product of two random ...
1
vote
0answers
72 views

Cesaro mean of products of converging matrices

Let $S$ be a finite set of states. Let $(M_n)$ be a sequence of transitions on $S$; that is, for every natural number $n$, $M_n$ is a non-negative $|S| \times |S|$ matrix whose rows sum up to 1. ...
3
votes
1answer
134 views

distribution discretization

Let $\mu$ be a distribution on $\mathbb{R}^n$. We partition $\mathbb{R}^n$ into small cubes congruent with $[0,\delta)^n$, parallel to the axes. In each cube, pick a point $x$ (for instance, the ...
2
votes
1answer
185 views

Almost sure convergence and weak star convergence

Let $(f_n)_{n\in \mathbb{N}}$ be a sequence of nonnegative measurable functions in $L_1[0,1]$. Assume that $$f_n \to f, a.e.$$ and $$\int f_n h \to \int g h, \forall h \in C[0,1].$$ Question: Do we ...
2
votes
1answer
108 views

Maximizer of random walk with very small drift

This is an extended question based on Large deviations for maximizer of random walk with drift. Let $$S_k = X_1 + \ldots + X_k,$$ where $X_i$ are i.i.d. with mean $-\mu < 0$ and unit variance. ...
1
vote
0answers
63 views

A sequence of Dirac Measure converging to stationary distribution of a Markov Chain

I have seen in a paper(http://www.sciencedirect.com/science/article/pii/S0167691105001040) Page 143 lemma 3.1, the following : Let $X_n$ be an ergodic markov chain taking values in a complete ...
2
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0answers
43 views

Non-autonomous O.D.E with discontinuous and not integrable R.H.S

Consider the non-autonomous O.D.E $\dot{x}(t) = \int h(x(t),y)\mu(t,dy)=F(x(t),t)$ where $\mu(t,dy) = \delta_{y_n}(dy)$ when $t \in [t_n,t_{n+1})$ and $h:\Bbb R^d \times S \to \Bbb R^d$ where ...
7
votes
4answers
288 views

Gaussian distributions as fixed points in Some distribution space

I'm taking a course on topology and probabily. Today, the professor remarked something along the lines of: If you look at the space of probability distributions with $0$ mean and variance $1$, ...
0
votes
2answers
111 views

What are some examples of isotrophic sets?

What are some examples of isotrophic sets? and is there a "good" way to describe them? Isotrophic meaning that a random vector X uniformly distributed in the set has the isotrophic property for all ...
3
votes
1answer
74 views

Could quadratic variation determine distribution?

Let $M=\{M_t,\mathcal{F}_t;0\le t<+\infty\}$, $N=\{N_t,\mathcal{F}_t;0\le t<+\infty\}$ be two continuous local martingales with $M_0=N_0=0\text{ a.s.}$. If $\langle M\rangle=\langle N\rangle$, ...
1
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0answers
88 views

Probability, Topology, functional analysis problem

Consider the coarsest topology on the space of functions, whose value at a point is a probability measure on a polish space $S$, s.t. the follwoing function is continuous : $$\nu(.) \to \int_{0}^{T} ...
6
votes
2answers
254 views

A version of Wald identity

Let $W$ be a standard one-dimensional Brownian motion. Let $T$ be a stopping time with $\mathbb{E}\sqrt{T}<+\infty$. Then $$\mathbb{E}W_T=0\quad \mathbb{E}W^2_T=\mathbb{E}T$$ I can prove these ...
0
votes
1answer
252 views

A question on Ito integral

Let $W$ be a standard one-dimensional Brownian motion and $0<T<+\infty$. Then $$\lim_{\beta\to+\infty}\sup_{0\le t\le T}|e^{-\beta t}\int_0^te^{\beta s}\mathrm{d}W_s|=0\quad \text{a.s.}$$ Could ...
2
votes
0answers
61 views

Existence of 1-1 mapping/homeomorphism

Let $B$ be a standard 2-D Brownian motion, and $\sigma: \Omega\times \mathbb R^{+} \mapsto \mathbb R^{2 \times 2}$ is an $\mathcal F_{t}$ adapted process satisfying, for some constants ...
8
votes
2answers
206 views

limiting distribution of the random walk from irrational rotation

Motivation: If I recall correctly, the simple symmetric random walk from i.i.d binary steps converges in distribution to the Wiener measure (if scaled with $a_n = \sqrt{n}$). What I am wondering is ...
3
votes
0answers
98 views

Properties of a finite random walk

Consider the simplest random walk - $X_0 = 0$ and from there on (i.i.d), $X_i=X_{i-1}+1$ with probability $p$ or $X_{i-1}-1$ otherwise. Let $Y_N$ be the highest point $X$ have reached on the first ...
8
votes
4answers
700 views

What does it mean when we say we have computed a number to a certain accuracy using a probabilistic algorithm?

My intention is to ask a general question about probabilistic (Monte Carlo) algorithms. But to keep things simple, I will focus on a few specific examples. Let me start the discussion with ...
2
votes
1answer
107 views

Algorithm interpolating between alternation and randomness

Is there an algorithm, not necessarily in the TCS sense, that is a canonical interpolation between alternation and randomness in sequences of binary digits? This hopefully illustrates my question: A ...
2
votes
2answers
198 views

Removing subtrees

Let $T$ be a complete infinite rooted binary tree. Is it possible to remove (infinitely many) subtrees of $T$ and get a subgraph $G$ such that: $G$ has no complete subtrees (the graph below any ...
5
votes
1answer
282 views

Can ergodic theory help to prove ergodicity of general Markov chain?

I am a beginner in ergodic theory. I have read some lecture notes(such as this and this) about it in hope that I could find something which helps to prove the ergodicity of some Markov chain taking ...
8
votes
1answer
352 views

Local time of Brownian motion + Lipschitz continuous function

Let $\mathrm{ Lip} (M)$ denote the space of all functions on $[0,T]$ with Lipschitz constant and $L^\infty$ norm bounded by $M$. Let $(B_t)_t$ be a Brownian motion defined on the probability space ...
0
votes
1answer
141 views

A conditional expectation question about consecutive inner products

Consider two random vectors $v=(v_1,\dots, v_n)$ and $w=(w_1,\dots, w_{n+1})$. Each $v_i \in \{-1,1\}$ independently and with equal probability. Each $w_j \in \{-1,0,1\}$ independently with equal ...
1
vote
0answers
109 views

How to show that two linear combinations of Bernoulli random variables have jointly Gaussian distribution (and more)

Let $X_1,\ldots,X_n$ be independent Bernoulli random variables such that $\mathbb{P}(X_i=\pm 1)=1/2$ and consider two collections of real numbers $a_1,\ldots,a_n, b_1,\ldots, b_n$. For the moment let ...
1
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0answers
57 views

limit distribution of multinomial distribution with increasing categories

If $\bf{X} \sim \text{multi}(n,p)$ with $k$ categories, we know $$ \sqrt{n}\left( \frac{\bf{X}}{n} - \bf{p} \right) \rightarrow^D N(0,\Sigma),$$ where $\bf{X}=(X_1,\ldots,X_k)^T$ and ...
3
votes
1answer
131 views

Moments of random special unitary matrices

This should be both well-known and probably easy, but I was wondering if the following is known (and, if so, how to easily calculate the thing or where to read about how to calculate it): what is ...
0
votes
0answers
60 views

Local limit theorem for an infinite dimensional integer lattice

Can someone refer me to a local limit theorem for the sum ${\bf S} = \sum_{i=1}^n{\bf X}_i$ of a sequence of independent and identically distributed $d$-dimensional random variables $\{{\bf ...
2
votes
0answers
63 views

Long paths in the supercritical percolation.

I have a question on the length of the longest path, denoted by $\ell_n$, in the supercritical percolation on $[0,n]^d$, denoted by $C_n$. We know that $C_n$ has a giant component whose size is of ...
-2
votes
1answer
173 views

using jensen's inequality

Suppose we have an expression f(x, h(x,y)), for some function f and h, and x, y are random variables, now we know that the function f(a, b) is concave w.r.t. a for given b. Can we use Jensen's ...
2
votes
1answer
129 views

Onsager-Machlup function and most probable path of a diffusion process

Let $X_{t}$ be a real, one-dimensional diffusion process satisfying the stochastic differential equation \begin{equation} dX_{t} = f(X_{t})dt + dW_{t}, \end{equation} where $f \in C_{b}^{2}(R)$ is a ...
0
votes
0answers
56 views

Question about the representation of Skorokhod

I have a question about Skorokhod's representation theroem. Let $\Omega=R^m$ and define the canonical process $X=(X_1, ..., X_m)$, i.e. $X(\omega):=\omega$ for any $\omega=(\omega_1,..., \omega_m)\in ...
-2
votes
2answers
81 views

Convergence of series made out of Markov Chain

$\{X_n\}$ be a ergodic Markov Chain taking values in $\Bbb Z$. Can I find some sufficient condition under which the $E[e^{\sum_{i} |X_i|}] < \infty$ (or say with some high probability).
1
vote
1answer
106 views

Large deviations for maximizer of random walk with drift

Is it easy to write down the large deviations rate for the maximizer of a random walk with negative drift? Let $X_i$ be the (iid, mean $-\mu$, variance $\sigma$, arbitrarily nice tails) jumps of a ...
1
vote
2answers
157 views

Proving a random bipartite graph contains a perfect matching

I have the following problem consider a random bipartite with vertex classes $A$ and $B$ of size $|A|=|B|=\mathrm{log}^{2}(n)$ graph in which every possible edge is chosen independently with ...
6
votes
3answers
385 views

Probability that a self-avoiding walk on $\mathbb{Z}^3$ closes to a polygon

The probability that a random walk on $\mathbb{Z}^3$ returns to the origin is about 34%. This is (part of) Pólya's theorem. I have been looking for an analogous (numerical) result for the probability ...
1
vote
1answer
130 views

Does very fast convergence in probability imply almost sur convergence for a continuous stochastic process?

I was wondering if someone knows how to prove the following fact (which might not be a fact ;) ): let X being a stochastic process with almost surely continuous sample path, and such that, there ...
1
vote
0answers
88 views

A natural sum over multisets (expectation over multinomial)

I think this is a natural question but am not sure where to find resources. Consider the possible multisets arising from choosing $n$ times an item from one of $k$ categories. We can represent one ...
-1
votes
1answer
115 views

Property of relative entropy [closed]

For $X$ a measurable space and $P,Q$ two probability measures on $X$ s.t. $Q$ is absolutely continuous with respect to $P$, the relative entropy is defined as $$D(Q\|P)=\int_X \log(\frac{dQ}{dP})dQ,$$ ...
2
votes
0answers
43 views

Local time for drifted Brownian motion and comparison results for reflected diffusion

Suppose $X(t) = x+ \mu t + \sigma W(t)$ where $x\ge 0$, $\mu, \sigma>0$ are real constants, and $W$ is a standard Brownian motion. The Skorohod decomposition of $X(t)$ can be written as $Z(t) = ...
0
votes
0answers
54 views

Computation on Random Bipartite graphs

I'm looking at a random bipartite graph $K_{\omega(n)}*K_{\omega(n)}$ where $\mathrm{log}(n)\leq \omega(n) \leq n^{1/2}$, in which each of the $\omega(n)^{2}$ edges is placed randomly with probability ...
5
votes
0answers
184 views

A generalization of Jensen's Inequality

Jensen's inequality is well known as $$E\big[f(X)\big]\le f\big(E[X]\big)$$ where $X$ is a integrable random variable and $f: R\to R$ is a bounded concave function, see also ...
10
votes
1answer
269 views

Does Brownian motion immediately visit both sides of a Jordan curve?

Let $C$ be a Jordan curve in $\mathbb{R}^2$. By the Jordan curve theorem, $\mathbb{R}^2 \smallsetminus C$ is uniquely partitioned into two connected regions $A$ and $B$ (the interior and exterior). ...
0
votes
1answer
84 views

Probability and Markov processes

Suppose I have a Markov chain (satisfying all conditions of ergodicity) that has a stationary distribution that is easy to sample from. ( Assume that we know the stationary distribution upto a ...
3
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0answers
65 views

Reasoning about dependent and independent quantities by “degrees of freedom”

In his classic textbook Foundations of the Theory of Probability Kolmogorov defines Independence a little bit differenent then it is usually done today. He denotes a probability space by $(E, \mathcal ...
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0answers
52 views

A series with long-tailed terms

Let's consider the following series: $$ \zeta = \sum_{k=1}^{\infty} a_k \xi_k, $$ where the sum is understood as the limit in $L_2(\Omega)$, $a_k \in \mathbb{R}$, $\sum_{k=1}^{\infty} a_k^2< ...
6
votes
2answers
121 views

Geometric interpretation of the average of two independent Cauchy distributions

Let me state two facts: (1) It is well known that if one takes a point uniformly distributed on the unit circle, and then takes it stereographic projection, the corresponding measure induced on the ...
0
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0answers
25 views

Coordinates Poisson Cluster parent point

Is there any method to know the position of parent point in 'Poisson Cluster Process'? For information I use data with poisson distribution. data consist of (longitude, latitude, date). I want ...
2
votes
1answer
107 views

Quaternion Wishart matrices of half-integer dimension?

For a physics application (quantum delay times of a chaotic scatterer) I need to generate $m$ positive random variables $\lambda_1,\lambda_2,\ldots\lambda_m$ with probability distribution ...
0
votes
1answer
69 views

Extend product sigma-algebra to cross-constant sets

We have two probability spaces $(\Omega_1,\mathcal{F_1},P_1)$ and $(\Omega_2,\mathcal{F_2},P_2)$. Is it possible to construct probability space $(\Omega=\Omega_1\times\Omega_2,\mathcal{F},P)$ such ...