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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299 views

Supremum in a Markov chain model

A Markov chain $X$ with finite state space $\{1,2,\cdots,N\}$ is defined on a probability space $(\Omega, P, \mathcal{F})$ equiped with filtration $\{\mathcal{F}_t\}$. And we assume that we can reach ...
0
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1answer
480 views

probability mass function fitting [on hold]

I have a probability mass function of some experimental data who's log looks like the following: (please ignore the fact that it is not normalized) ![alt text][1] [image shack image removed] ...
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1answer
356 views

Size of KL-divergence neighbourhoods

I am new here. I was reading another post here and this got me wondering what can be said about the size of the following kl divergence neighborhoods. Consider these two kl-divergence neighbourhood ...
4
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1answer
76 views

Time for brownian motion to cross a coordinate plane

Can I get a reference or some insight into the following? Suppose a particle moves by Brownian motion, starting from a point $P$ in $\mathbf{R}^{n}$. What can we say about the distribution of the ...
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0answers
119 views

Pointfree probability theory

I must confess I hardly know anything about probability theory. Still, I'm interested in the following: Much like pointfree topology, where one basically replaces topological spaces by their locales ...
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0answers
28 views

Bounding Expected Value of a piecewise function [on hold]

Let X and Y to be two independent random variables with known pdfs. Get a bound for the expected value of the following expression in terms of $E[X]$, $E[Y]$, VAR[X] and VAR[Y]: \begin{equation} ...
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2answers
111 views

Reference to iterated logarithm law and Smirnov law of empirical CDF

I am reading V. Vapnik's "Statistical Learning Theory". The author layouts following two statistical laws related to empirical CDF. I am looking for reference about proofs on these two laws. Let ...
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1answer
143 views

Matching moments in even dimensions

Let $D$ be a probability distribution on the unit interval $[0,1]$ with moments $\mu_i=\mathbb{E}_D [x^i]$. Let $\delta(x)$ be a singleton probability distribution with all weight at $x\in [0,1]$. ...
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0answers
118 views

Tensorization of Orlicz norm?

Associated with a convex function $\phi:[0,\infty)\mapsto[0,\infty)$ satisfying $\lim_{x\to 0} \frac{\phi(x)}{x} = 0, \lim_{x\to\infty}\frac{\phi(x)}{x} = \infty,$ the Orlicz norm of a random variable ...
3
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1answer
105 views

Practical bounds for the Wasserstein distance in 2 dimensions

Let $X_1,\dots,X_n$ be a set of independent samples of a distribution $\mu$ on the unit square, let $\hat\mu_n$ be the empirical distribution on the points $X_1,\dots,X_n$, and let ...
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0answers
77 views

Convergence rate of Pearson correlation matrix

I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let ...
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0answers
40 views

Coupling Marginals of Distributions on the Sphere

Given a distribution $P_X$ on $\mathbb{R}$, when does there exist a coupling (i.e. joint distribution) $P_{X^n}$ of $X_1,...,X_n$, each distributed according to $P_X$, such that $\sum X_i^2 = n$ ...
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1answer
74 views

Orthogonal polynomials with respect to the lognormal distribution

I am currently doing some inspection on the orthogonal polynomials with respect to the lognormal distribution. Does anyone already work on that or know some cool references? All the best, Pierre-O.
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1answer
151 views

$\langle X\rangle_t = t$

Suppose $B_t$ is a standard Brownian motion in $\mathbb{R}^d$ and $X_t = |B_t|$. What is the easiest way to see that$$\langle X\rangle_t = t?$$I need this result for a simulation I am running...
3
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1answer
68 views

Error for the convergence by distribution

A sequence of random variables $X_n$ converges in distribution to $X$, if there is pointwise convergence of its characteristic functions, i.e. $\lim_{n\rightarrow\infty}\phi_{X_n}(\lambda) = ...
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1answer
126 views

inequality with exponents

We are given a graph $G$, each vertex $v$ has an assigned value $\gamma_v\in [0,1]$, and it happens that for every $v$ we have $\gamma_v+\sum_{u\in \delta(v)} \gamma_u = 1$. Assume that $\sum_v ...
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0answers
20 views

Statistics, the deviation and expection of a number sequence [closed]

There is a sequence of number $a_{0},a_{1},...,a_{n}$, $(0 < a_{i} < 1)$ Define $b_{t} = \frac{ \sum_{i=0}^{t}{w^{t-i}a_{i}} }{ \sum_{i=0}^{t}{w^{t-i}} }$ where $w \in (0, 1)$. Can we proof ...
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62 views

Probability two random intervals overlap

I'm working on an algorithm for orthogonal line intersection detection and am trying to analyze some things about it. For simplicity, we can consider the problem as follows: Given N randomly ...
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61 views

Tail bound for a martingale

The setup is as follows. We are given a martingale $X_0,X_1,...,X_k$. The difference $X_i-X_{i-1}$ is always between $[-1,1]$. Variance $D^2(X_i-X_{i-1}| X_{i-1})$ is something, but we can show that ...
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0answers
53 views

Probability and Statistics [closed]

for the cards shown below, what is the probability of choosing a yellow card and then a D if the first card is replaced before the second card is drawn? [b] [1] [5] [D] [10] ...
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97 views

Involutions on $[0,1]$ given by power series (related to probability generating functions)

Let $A$ be a function from $[0,1]$ to $[0,1]$. $A$ is an involution if $A(A(x))=x$ for all $x\in[0,1]$. Which involutions $A$ exist such that $A(x)=\sum_{k=0}^\infty a_k x^k$ with $a_0=1$ and ...
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66 views

Uniqueness of the “Gubinelli” Derivative in the Theory of Paracontrolled Distributions

From the theory of Rough Paths it is well known that if we have a truly rough path $X$ and two controlled rough paths $(Y,Y'),(Y,\tilde{Y}')\in\mathcal{D}_X^{2\alpha}$, then we have already $Y' = ...
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1answer
313 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 ...
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3answers
945 views

Persistent homology of Gaussian Fields in Euclidean space

If you generate points in $\mathbb R^n$ via a process that respects a Gaussian normal distribution, then compute the persistent homology / barcodes, to my eye something fairly regular seems to be ...
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1answer
596 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 ...
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2answers
625 views

How should a mathematician approach the physics literature concerning percolation?

I would like to read some of the physics literature on two-dimensional percolation, however in attempting this I have run into two problems. (1) Physics papers on percolation are (relatively) hard ...
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5answers
438 views

Identities and inequalities in analysis and probability

Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of ...
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415 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 ...
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0answers
94 views

Do binary symmetric channels maximize mutual information?

Consider the following setup: $(X, Y)$ is a doubly symmetric binary source with parameter $0 < p < 1/2$, i.e., $X \sim \text{Bernoulli}(1/2)$, $Z \sim \text{Bernoulli}(p)$ and $Y = X \oplus Z$. ...
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431 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 ...
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35 views

Quotient of cumulative binomial distribution functions

Given to integers $n < m \in \mathbb{N}_0$ and a probability $p$, I'm struggling to calculate (or at least get an upper bound for) the quotient $$Q = \frac{F(n+1;m,p)}{F(n;m,p)}$$ where $F$ denotes ...
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33 views

Having the highest value in a interval appear less often [closed]

I have an array of size 5. And initially in each index, they are initialized with the value 1. so it looks like this : 1 1 1 1 1 Every iteration, I get a decimal value between 0.0 and 1.0 At the ...
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1answer
816 views

Hardy spaces: analysis <---> martingales

Let $H^p$ be the Hardy space of analytic functions on the open unit disk $\mathbb{D}$: $f \in H^p$ if $f$ is analytic on $\mathbb{D}$ and $\sup_{r < 1} \int_0^{2\pi} |f(re^{i\theta})|^p d\theta ...
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0answers
236 views

A variant frobenius problem

From Sylvester's theorem we know that using only coins of sizes $a,b$, we can change exactly $\frac{(a-1)(b-1)}2$ different big coins up to $(a-1)(b-1)$. Denote sets ...
4
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1answer
278 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 ...
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0answers
95 views

$\mathbb{P}(d(X,Y)>\alpha)<\beta$ if $\mathbb{P}(X\in E)\leq \mathbb{P}(Y\in E^{\alpha})+\beta$ for all measurable E

Given two random variables X,Y with measures P,Q. Show that if $P(E) \le Q(E^\alpha) + \beta$ for all measurable $E\subset\mathbb{R}$ then $\mathbb{P}(d(X,Y)>\alpha)<\beta$. Only hints please. ...
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39 views

Convergence of approximate quadratic variation in $L^p$

For a semimartingale $X_t$, I can set $$[X]^N_t = \sum_{j=1}^N \bigl(X_{t\frac{j}{N}}-X_{t\frac{j-1}{N}}\bigr)^2$$ Then it is well-known that the process $[X]^N_t$ tends to the quadratic variation ...
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1answer
106 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 ...
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0answers
24 views

Derivative of a cdf with respect to a parameter

Given two independent Random Variables $X$ and $Y$ with known distributions, I would like to know if I can say that the expression $$ \operatorname{Pr}( f (t'+Y-X)+Y-X < z) $$ is increasing in ...
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1answer
525 views

Is the set of the convolutions of two-point measures dense in the set of all measures?

A measure supported in two points is a measure of the form $$ \mu=\alpha\delta_a+(1-\alpha)\delta_b, $$ where $a<b$ and $\alpha\in (0,1)$. The question is: Given a finite non-negative measure ...
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85 views

A generalization of coupon collector problem - $\geq1$ pick per experiment

Mix $T\geq1$ coupons numbered $1$ to $T$ with a set of $S\geq0$ number of dummy coupons with no numbers. Select $N\geq1$ coupons at each trial at random and put them back. $N=1$ is standard coupon ...
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47 views

Compute the Gibbs energy

I have a question about Gibbs distribution in Stochastic theory. In which, it defined a clique as a a subset $C$ in the whole image $\Omega$ if two different element of $C$ are neighbors. Figure 2 ...
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1answer
242 views

Existence of Limiting Distribution for Moving Regions in Stat. Phys. Models

As the title (hopefully) suggests, I've been trying to prove (or disprove) the existence of a limiting distribution for a certain projection in a statistical physics model. I'll give the details of ...
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3answers
382 views

Expectation of $(c+e^{N(0,\sigma^2)})^{-n},\, n>0$

I would like to know if there's a way to compute or approximate the following expectation: $$\mathbb{E}[(c+e^X)^{-n}]$$ where $X=N(0,\sigma^2)$ and $n,c>0$ (you can also assume that $n$ is a ...
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2answers
1k views

Variance of exponential random variable

For a random variable $\xi$, what bounds can be achieved for Var $e^{\xi}$ in terms of E$\xi$ and Var $\xi$?
5
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1answer
317 views

Stochastic process describing long-term fluctuations

I need to model a process that has large, smooth and mean-reverting long-term fluctuations and some small short term wiggles, a sample path looks like this: My first idea was to model it as an ...
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1answer
198 views

Can we estimate the probability $\mathbf{P}(a-k|a - b) $ on a random graph?

Let $G=(V,E)$ be an undirected random graph such that $V$ is the set of nodes, and $E$ is the set of edges Assume the ground graph $G$ is sparse enough, for example, $\frac{|E|}{|V|}= c \in [10, ...
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1answer
701 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 ...
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73 views

Finding a hidden “heavy” subset of random variables

Let $X_1,\dots, X_n$ be independent non-negative random variables (with finite expectation and variance), and $0 < m < n$ be a fixed integer such that there exists a subset $S\subseteq [n]$ of ...
5
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2answers
209 views

General additive function of probability

Let $H$ be a function of finite sequences of probabilities (non-negative numbers summing up to 1) into real numbers, such that: $H$ is continuous, $H$ is symmetric w.r.t. the order of its arguments, ...