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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2
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62 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
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0answers
42 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
48 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 ...
7
votes
0answers
101 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
67 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
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4answers
636 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
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 ...
-4
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0answers
31 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 ...
2
votes
2answers
176 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 ...
-2
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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 ...
5
votes
1answer
228 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 ...
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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 ...
7
votes
1answer
225 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
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1answer
104 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
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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 ...
1
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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 ...
3
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1answer
103 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
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0answers
52 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
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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
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 ...
-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 ...
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0answers
49 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 ...
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votes
2answers
70 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
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 ...
1
vote
2answers
136 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
366 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
114 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 ...
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vote
0answers
75 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 ...
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votes
1answer
105 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
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0answers
37 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
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0answers
48 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
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0answers
153 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
265 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). ...
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0answers
51 views

Establishing CDF of sup of Brownian motion and Brownian Bridge

Question 1: Let $W_t$ be a Brownian motion. Then how could we prove that $$\Pr\left\{\sup_t|W_t|<b\right\}=1-\frac{4}{\pi}\sum_{j=1}^\infty \frac{(-1)^j}{2j+1} ...
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0answers
23 views

Non homogeneous poisson process

I'm trying to model a chemical reaction using a poisson process but with a little tweaking. I want a rate $\lambda$ that depends on $X_t$ which is the quantity of one of the chemical compounds. For ...
0
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1answer
80 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
votes
0answers
61 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
50 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
104 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
votes
0answers
24 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 ...
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0answers
27 views

Expectation of a function with a Gamma distributed random variable

Consider a truncated exponential distribution $F(x\left| \lambda \right.) = \frac{{ - {e^{ - \lambda x}} + {e^{ - \lambda }}}}{{ - {e^{ - 2x}} + {e^{ - \lambda }}}}$ on the interval $[1,2]$. The ...
2
votes
1answer
98 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
63 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 ...
2
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0answers
79 views

Probability question involving simulations of picking balls from a bag

I’m working on a chemistry problem, which essentially translates to finding the answer to a related probability problem. However, my knowledge in probability is very limited and I'd be grateful if ...
0
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0answers
75 views

What is the sigma field of the derivative of a process?

When $t\to X_t$ is an absolutely continuous process ($X_t= X_0+ \int_0^t Y_s dt$ for some measurable process $Y_t$) we have for all $t$ $$\sigma(Y_t) \subset \cap_{\epsilon >0}\sigma(X_{s}, s\in ...
10
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2answers
468 views

Teaching stochastic calculus to students who know no measure theory (or PDE, or…)

I've got quite a challenge as my teaching assignment for the next Fall (not that I want to get rid of it, quite the contrary, but I still feel like asking for advice won't hurt :-)). I'm to teach the ...
0
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0answers
46 views

Central limit theorems for unequal probability sampling (weak but ill-defined dependence)

Suppose we are choosing samples of size $s$ from a finite population $\{a_1, a_2, \dots , a_n\}$ where our sampling is with unequal probabilities. Construct $$ S_n = \sum_{k=1}^{n} a_k $$ Under what ...
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0answers
78 views

Defining density of a random function using Radon-Nikodym Theorem

Let $(\Omega,\mathbb{F},P)$ be a probability space and $E$ be an infinite dimensional Banach space and $\mathbb{B}$ be the $\sigma$-algebra of Borel subset of $E$. Let $X$ be random function defined ...
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0answers
52 views

When the completed filtration of a process increases slowly

If $\mathcal{F}_t$ is the filtration of the evaluation process on $C_T$ (continuous function on $[0,T]$). Can we find some law of continuous process $\mathbb{P}$ so that for $t\leq T$ ...