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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3
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61 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
91 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
72 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
45 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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0answers
61 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 ...
-3
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0answers
87 views

Camel up Board game probability problem [closed]

There is this game called "Camel Up" which is basically about betting on camels. I wanted to calculate probabilities that every camel has to end up 1st after 1 turn AND probability that every camel ...
-1
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0answers
32 views

Equality in fraction of density [on hold]

For two densities $f_1,f_2$ which take value in $[t_{min},t_{max}]$ following equality holds $$\frac{f_1(x+j)}{f_1(x)}=\frac{f_2(x+j)}{f_2(x)}$$ for all $j\in\mathbb{Z}$ and all $x\in\mathbb{R}$, ...
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0answers
49 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
59 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 ...
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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 ...
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1answer
156 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 ...
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0answers
31 views

Batch Markov Arrival Process - Computing Ps(t) [closed]

Suppose to have a queue $Q$ that represents a finite size buffer. We have multiple arrivals to the queue with the same arrival rate $\alpha$. Every group that comes to the queue can have a maximum ...
2
votes
1answer
105 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 ...
-3
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0answers
30 views

an exercise about conditional expectation [closed]

Consider the independent random variables X and Y are in exponential distribution with parameter 1( it means P(X>x)=exp(-x) for all x>0 and n>=0). Determine: (a) E (X | X + Y); (B) E (X | max (X, 1)). ...
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0answers
25 views

question: a case of martingale stopping theorem application [closed]

Consider the following steps on pairs of integers (m, n) for m, n ≥0: we start at (0,0) and then we go from (m, n) to (m + 1, n) or (M, n + 1) with probability 1/2 for each case. Γ is a sequence of ...
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0answers
21 views

Martingale stopping theorem [closed]

English: By using the martingale stopping theorem, determine the expectation of the number of characters that a child will eventually spell the postal code of the father Noel H0H 0H0, assuming he ...
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0answers
47 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
67 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
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1answer
84 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
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2answers
134 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
363 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
110 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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0answers
73 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 ...
0
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1answer
103 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
34 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) = ...
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0answers
60 views

density of penalizations of Gaussian probability measures

Let $\mu$ be a Borel probability measure on $\mathbb{R}^d$. By following one of the standard proofs of Bochner's Theorem (mollify, use Fourier inversion and Levy's Continuity Theorem), it is easy to ...
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
137 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
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1answer
260 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
48 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} ...
-1
votes
0answers
21 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
76 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
59 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
49 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
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2answers
97 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 ...
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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
95 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
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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
75 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
73 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 ...
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2answers
444 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 ...
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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
76 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
51 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 $t\leq T$ ...
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0answers
79 views

Estimating the moments of a random variable

Suppose i wanted to estimate the expectation and variance of a random variable $X$. More over suppose i could write a variable $X$ as a sum of indicator random variables $X=\sum_{i=1}^{k} X_{i}$. Are ...
2
votes
1answer
58 views

Estimating mean and variance of a distribution based on error-prone estimates of its cdf

Suppose I have some random variable $X$ taking values in $[a, b]$ with unknown distribution (I am happy to assume the distribution is smooth, though it would be nice to not have to). I have a ...
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0answers
88 views

Worst-Case Solution to (Stochastic) Matrix Inequality

EDIT: Some specific conjectures added. This problem comes with an associated stochastic process, but I phrase everything as linear algebra in case somebody from a non-probability community has seen ...
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0answers
42 views

Reference for a special case of the Hanson-Wright inequality

I would like find tail bounds for the expression $$ \begin{align*} \left|\left\langle a,\phi\right\rangle \left\langle \phi,b\right\rangle -\left\langle a,b\right\rangle\right|, \end{align*} $$ where ...
8
votes
3answers
285 views

Reference for a strong intermediate value theorem for measures

Let $\mu$ be a finite nonatomic measure on a measurable space $(X,\Sigma)$, and for simplicity assume that $\mu(X) = 1$. There is a well-known "intermediate value theorem" of Sierpiński that states ...