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
votes
3answers
618 views

Intuitive explanation of Dvoretzky's theorem

I am wondering if anyone has an enlightening explanation of why Dvoretzky's theorem (which says that a high-dimensional convex body has an almost round central section) is true -- there are a number ...
7
votes
1answer
178 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 ...
2
votes
0answers
42 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 ...
4
votes
1answer
336 views

Strong Law of Large Numbers for arrays of partly dependent random variables

Suppose $X_1$, $X_2$ are two independent real-valued random variables. Let $F$ be a continuous (unbounded) function from $\mathbb{R^2}$ to $\mathbb{R}$. Assume that the necessary measurability and ...
0
votes
1answer
386 views

Probability generating function zero implies random variable is infinite

Let $V$ be a random variable supported on the nonnegative integers (including $\infty$) and $f(x) = \mathbf E x^V$ be the probability generating function. In our model $V$ is the number of visits to ...
5
votes
0answers
72 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 ...
-2
votes
0answers
51 views

Blood type frequency given probability [migrated]

I have calculated the probability that any child will have a particular blood type from both the genotype level and the phenotype level assuming the human ABO Rh system is followed. Here are the ...
2
votes
0answers
54 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
633 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 ...
4
votes
1answer
279 views

Law of Iterated Logarithm for autoregressive process

Suppose that $\{X_i\}$ is an $\mathrm{AR}(r)$, defined by: $X_{i}= h(i) + \varepsilon_i $, $h(i)=\alpha_1 X_{i-1} + \dots + \alpha_{r} X_{i-r}$ where $\{\varepsilon_i\}$ are i.i.d. ${\cal ...
2
votes
1answer
242 views

Derive concentration bound for the derivative

It that true to conclude that if a random $f(z)$ is a sub-Gaussian random variable for a constant value of z, its derivative $f'(z)|_{z=k}$ with respect to variable $z$ is also sub-Gaussian? In ...
3
votes
1answer
100 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 ...
2
votes
1answer
99 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
1answer
172 views

Quasi-stationary distribution for a death process

In the paper, Survival in a quasi-death process by van Doorn and Pollett, the quasi-stationary distribution of a transient CTMC is discussed and QSD for a simple death process is derived. Consider a ...
5
votes
1answer
280 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 ...
-4
votes
0answers
30 views

Probably some naive question on conditional probability [closed]

As known, three variables x_1, x_2 and y, if x_1 and x_2 are conditional independent given y, we have p(x_1, x_2|y) = p(x_1|y)p(x_2|y). I was wondering about p(y|x_1, x_2), is that possible to get ...
0
votes
1answer
128 views

Poisson approximation of random sub-graphs

I add the edges of $G(n)$ the complete graph on $n$ vertices one by one, at random and without replacement, and denote by $G(n,m)$ the resulting Erdos Renyi random graph process. At step $m$ in the ...
2
votes
2answers
174 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 ...
3
votes
2answers
328 views

trivial map on $\sigma-$algebra $\mod{}0$ is trivial

Hi everyone! I am currently studying the basic theory of measurable actions and need the following result, which I am not able to prove myself. It is stated without a proof, so probably it should not ...
5
votes
1answer
295 views

Table with the most seated customers in Chinese restaurant process

Suppose we have some initial configuration of people seated at some tables. We start taking new customers and seat them following Chinese restaurant process. Is there some known work on finding the ...
5
votes
1answer
226 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 ...
2
votes
1answer
177 views

weak convergence of the solutions to stochastic heat equation

$W(t,x)=\sum_ic_ie_i(x)B^i_t$ is a Brownian motion in $L^2(R^d)$, where $\{e_i\}$ is the standard orthogonal basis and $\sum_ic_i^2<\infty$. $$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$ ...
8
votes
1answer
425 views

Joint law of the time integral of Brownian motion and its maximum

Suppose $W_t$ is a standard one dimensional Brownian motion. Let $M_t$ and $I_t$ be its running maximum and time integral, respectively: $$M_t=\max_{0\leq s\leq t}\,W_s$$ ...
-2
votes
0answers
61 views

Finding an example for [closed]

Let $\varphi$ be a periodic function s.t. at zero and every integer points it is equal to 1. Moreover it's equal to one in at least one point between each integer. Can we have two distinct density ...
14
votes
2answers
687 views

self-avoidance time of random walk

How many steps on average does a simple random walk in the plane take before it visits a vertex it's visited before? If an exact formula does not exist (as seems likely), then I'm interested in good ...
3
votes
2answers
219 views

Expectation of a generalization of Dirichlet distribution

For the standard Dirichlet, the expectation of $X_i$ is $\alpha_i/\alpha_0$, where $\alpha_0 = \sum_i \alpha_i$ [http://en.wikipedia.org/wiki/Dirichlet_distribution]. I am considering the following ...
0
votes
2answers
103 views

Probability spaces involved in using Bayesian Inference

I am currently reading "Statistical and Inductive Inference by Minimum Message Length" by C.S. Wallace. In this, Wallace gives a fairly informal account of Bayesian Inference which, in the case ...
0
votes
1answer
425 views

For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?

I am trying to characterize all measures on $\mathbb{R}$ such that $$ \sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty, $$ where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...
8
votes
2answers
944 views

Probability of Generating a Connected Graph

$N$ points are generated randomly within a unit square, with a uniform distribution. What is the probability that the points form a connected graph, given that two points are connected if the distance ...
3
votes
1answer
168 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 \, ...
2
votes
1answer
213 views

Is there any result for upper bounding the tail of a sum of r.v.s by another tail (with a different threshold)?

Suppose $X_i$s are independent random variables. We can make assumptions about $X_i$, e.g., $X_i\in [0,1]$. Let $X=\sum_i X_i$ and $u=E[X]$. Is there any result of the following type (relating one ...
2
votes
2answers
107 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 $F$ ...
11
votes
1answer
566 views

Is there a probability theory developed in intuitionistic logic?

Since Boole it is known that probability theory is closely related to logic. According to the axioms of Kolmogorov, probability theory is formulated with a (normalized) probability measure ...
1
vote
0answers
80 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 ...
7
votes
1answer
326 views

Expected maximum inner product

If you sample $n$ vectors each with $m$ entries, with each entry chosen from the set $\{-1, 1\}$, how can you calculate the expected maximum absolute value of the inner product between all pairs of ...
5
votes
2answers
296 views

A central limit theorem for a trigonometric series involving primes

In some recent work I found I needed to prove a central limit theorem for the interesting series: $\sum_{n=1}^\infty \cos (u \log p_n) $ where u is a random variable uniform on the interval ...
0
votes
1answer
103 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 ...
20
votes
5answers
3k views

Probability of a Random Walk crossing a straight line

Let $(S_n)\_{n=1}^{\infty}$ be a standard random walk with $S_n = \sum_{i=1}^n X_i$ and $\mathbb{P}(X_i = \pm 1) = \frac{1}{2}$. Let $\alpha \in \mathbb{R}$ be some constant. I would like to know the ...
3
votes
1answer
232 views

Non-asymptotic large deviations for a convex set

Let $X_1,\dots,X_n$ be $n$ i.i.d random variables taking values in a Polish vector space $\mathcal{X}$ and with (Borel) probability distribution $\mu$. For any convex, compact $\Gamma \subset ...
1
vote
0answers
48 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 ...
8
votes
2answers
543 views

Sampling from Sine Kernel and Airy Kernel

A determinantal process on the line is a random collection of points on $\mathbb{R}$ such that the probability of $x_1, \dots, x_n$ lying on the random set is $\det (K(x_i, x_j))_{(i,j)}$. Examples ...
0
votes
1answer
537 views

A generalized urn-ball matching problem; Complicated combinatoric/probabilistic limit

I'm looking for a generalization to the urn-ball matching problem. As a reminder of what I've got in mind, here's the simple version: Randomly assign (with replacement) $N$ balls to $M$ urns. ...
0
votes
0answers
51 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
61 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 ...
1
vote
1answer
86 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 ...
-2
votes
1answer
161 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
114 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 ...
-1
votes
0answers
16 views

Probability of guessing the colors of a deck of cards correctly [migrated]

10 years ago when I was about 15 I sat down with a deck of shuffled cards and tried to guess if the next card in the deck would be red or black. In sequence I guessed 36 cards correctly as red or ...
1
vote
1answer
109 views

Diffusion processes with different diffusion coefficients and absolute continuity

I would first of all like to say that I am an analyst, and so I am familiar with probabilistic methods only on a basic level. My initial situation is the following. Consider two stochastic ...
2
votes
1answer
187 views

Mutual information decrease with coarse-graining

Let $X,A,Y,B,C,D$ be random binary variables. $D$ is independent from $X,A,C$ and $C$ is independent from $Y,B,D$. Is it true that: If $I(Y:B|D=0)\leq \epsilon$ then $I(X\oplus Y:A\oplus ...