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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65 views

When an integral with respect to a Poisson point process is finite?

Let $N(ds,dv)$ be a Poisson measure on $\mathbb{R} _+ \times \mathbb{R} _+$ with intensity $dsdv$. Let $N = \sum\limits \delta_{(s_i,v_i)}$. Assume that $N$ is compatible with a filtration $\{ ...
0
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1answer
87 views

When does a d.r.v. take a value very close to the mean? [on hold]

Suppose that $X$ is a discrete random variable with values $x_{1},x_{2},\ldots,x_{n}$ (not known precisely, but there is some information available about the mean and variance). Is there a result ...
6
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4answers
300 views

Expected value of a function over random sets

I am doing an analysis on the complexity of some set-related algorithm where the input is a random set. One of my setbacks can be formulated as follows: Pick $k$ distinct numbers out of numbers ...
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0answers
64 views

Bound on the sum of arguments

Problem: Show that for all real $s,t,u$ and all complex $z$ with $|z|<1$ one has $$(*)\qquad \arg\frac{1-zf(s-u)}{1-zf(s+u)} +\arg\frac{1-zf(t+u)}{1-zf(t-u)}<\pi, $$ where $f$ is the ...
2
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0answers
49 views

Nonlinear things that one can do to a probability density function [migrated]

Say $f(x)$ is a smooth probability density function on $\mathbb{R}^n$ with compact support region. This wikipedia page http://en.wikipedia.org/wiki/Maximum_entropy_probability_distribution explains ...
2
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0answers
53 views

Concentration bound in high min entropy distribution

Let $(X_{1},\dots,X_{m})$ be joint distribution on $\{0,1\}^{m}$ with that $H_{\infty}(X_{1},\cdots,X_{m})\geq m-r$, where $H_{\infty}$ means min-entropy. Let $P_{1},...,P_{n}\subseteq [m]$ be sets ...
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0answers
28 views

How to prove an inequality $\left| {g(j + 1)} \right| \le 5/4$ in Stein's method for Poisson approximation [migrated]

The following is a lemma in Barbour, A. D., Holst, L., & Janson, S. (1992). Poisson approximation. Oxford: Clarendon Press,p7. For $j=1,2,...$ and $\lambda > 0$, we have $\left| {g(j + ...
5
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1answer
111 views

Rademacher average based Hoeffding Inequality

I am following these lecture notes: Given the i.i.d. $\mathcal{Z}$-valued random variables $Z_1,\dotsc,Z_m$ and $\mathcal{G}$ is a set of bounded functions $g\colon \mathcal{Z}\to[a,b]$. Corollary ...
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0answers
34 views

Finding a random variable with a density function [on hold]

So I have this homework I'm having a really hard time starting: For the random variable X with density function f(x) = 4x , 0 < x ≤ 1/2 4 − 4x , 1/2 < x ≤ 1 0 , otherwise Determine the ...
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0answers
39 views

Shift invariance for the distribution of quadratic polynomials

For a probability distribution $X$, supported on integers, define the shift-invariance of $X$, denoted by $shift(X)$ = total variation distance between the random variable $X$ and $X+1$. Let ...
5
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1answer
159 views

Does independence of the sequence $f(A_i, B)$ imply the sequence is independent of $B$?

Suppose $B, \{A_i: i \in \omega\}$ are i.i.d. random variables with uniform distributions on $[0,1]$. If $f$ is a map such that $\{f(A_i, B): i \in \omega\}$ are independent, must $\{f(A_i, B): i \in ...
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0answers
38 views

Conditions on probability measure that generates non-void random polytope

Let $C$ be a non-void compact convex set in $\mathbb{R}^d$, and $\nu$ a probability measure on $C$. Then under what conditions on $C$ and $\nu$, the following statement is true: If ...
2
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0answers
114 views

Ticket lottery — distributing $n$ tickets among $N$ people fairly

Suppose that I have $n$ tickets for an event that I want to distribute fairly among $N > n$ people. In this simple case, a lottery suffices. But suppose certain groups of people want to attend ...
-2
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0answers
22 views

Expected values of non-negative random variables [migrated]

I met a problem during my research in computer science. I just want to know wether there is a relationship between E[X] and Pr{X>x}? E[x] = integral of pr{X>x} from 0 to infinite? But how to prove ...
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0answers
51 views

Expected value of a stochastic integral expression

I am wondering if the following expression can be processed a bit analytically, $$ E \left[ e^{aX} \int_0^X e^{bu}dW(u)\right], $$ where $W_u$ is the normal Brownian motion (1D Wiener process), and ...
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0answers
40 views

Taking power of the integrand in a Riemann-Stieltjie Integral

This is a problem I am trying to solve as part of a calculation for Value-at-Risk. Given that $P(X<x)=F(x)=\int_{\theta}F(x|\theta)dG(\theta)=1-\alpha$, where $F$ and $G$ are CDF's, is there a ...
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0answers
76 views

Proof of $\lim_{t\rightarrow 0} \mathbb E f(S_t)=f(0)$ for a diffusion $S_t$?

I am trying to prove the following statement for a diffusion $S_t$ with $S_0=0$ and a real function $f$ that is continuous at $0$: $$\lim_{t\rightarrow 0} \mathbb E f(S_t) = f(0), \text{ if } \mathbb ...
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0answers
52 views

Derivative of the Expectation of an Integral over a Diffusion

I am trying to prove the following, where $S_t$ is a diffusion: $$ \lim_{t\rightarrow 0}\frac 1 t \mathbb E \int_0^t f(S_s)ds = \lim_{t\rightarrow 0} \mathbb E f(S_t) $$ Proof attempt: Lusin's ...
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0answers
48 views

Consistency Conditions of the Kolmogorov Extension Theorem

Kolmogorov's extension theorem allows for the construction of a variety of measures on infinite-dimensional spaces, and its conditions are supposedly "trivially satisfied by any stochastic process". ...
5
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1answer
123 views

Asymptotic behavior of $X_n$ in a Dirichlet vector $(X_1, …, X_n)$

Let $(\alpha_k)$ be a sequence of positive numbers and let $(Y_k)$ be a sequence of independent random variables $Y_k \sim \text{Gamma}(\alpha_k,1)$. Set $X_n=\dfrac{Y_n}{\sum_{i=1}^nY_i}$. (edit) ...
3
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2answers
189 views

Picking codewords that are close

I posted this question in http://math.stackexchange.com/questions/1142698/picking-codewords-that-are-close a week back. Let $[n,k,d]$ be a linear code over $\Bbb F_q$ with minimum distance $d$ and ...
2
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0answers
102 views

Inequality with CDF of order statistics

here is a problem I have been struggling with for a while now. This is for a paper I am working on. Any help would be appreciated! Here we go: Each bidder's valuation $\theta _{i},$ $i=1,...,N$, is ...
2
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1answer
111 views

Does $\int \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \to \Phi(u)$ imply that $f_t \to \delta_1$?

I'm looking at a family $(f_t)$ of densities of some continuous random variables and know that $$\int_{-\infty}^{\infty} \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \xrightarrow{t \to ...
0
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1answer
47 views

Stationary distribution of random walk alias solving uncountably many linear equations [closed]

Let us have interval $I = (i_1,i_2)$, function $f_1 : I \mapsto I$, function $f_2 : I \mapsto I$. Let $x_0$, $x_1$, $x_2$, ... be series of random variables from interval $I$ denoting random walk. ...
5
votes
2answers
300 views

Applications of cohomology to probability and statistics

Are there interesting/useful applications of cohomology (and homological algebra in general) to probability and statistics, or information theory? By "interesting/useful", I mean "not merely ...
3
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0answers
46 views

Sample based inversion of the Radon transform

I have a classic tomography problem in which I would like to infer the internal density $p_0: \mathbb R^2 \to \mathbb R$ from external Radon projections. The internal density however is viewed as a ...
4
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0answers
60 views

“One sided” fast times of Brownian motion

Let $B_t$, $t \in [0,1]$ be a standard Brownian motion. We call a time $t$ fast up if $$ \limsup_{h \searrow 0} \frac{B(t+h) - B(t)}{\sqrt{2 h \ln(1/h)}} =1. $$ (Note the absence of absolute value ...
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0answers
30 views

Implementing the Bivariate survival function (Dabrowska estimator etc) [migrated]

I am trying to implement the Dabrowska estimator to apply to some data that has been collected and I am wondering if anyone can help. The original paper is Kaplan-Meier Estimate on the Plane by Dorota ...
0
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0answers
35 views

minimal entropy approximation of a truncated discrete measure

Consider a measure $\mu$ on $\mathbb{N}$ given by the sequence $(\mu(n))_{n \geq 0}$ with $\mu(0)>0$. For example $\mu(n)=n^2+1$ on the figure below. For each $n$, let $X_n \sim \mu(\cdot \mid ...
3
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2answers
231 views

Weak convergence of random measures

Let $\mu_n,n\in \mathbb N$ be a random probability measures and let $\mu$ be a deterministic probability measure on $\mathbb R$. That is to say, that the $\mu_n$ are measurable maps from a probability ...
2
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0answers
81 views

Equivalence of two non-degenerate Gaussian measures on Banach space

The motivation of this question is to show that two probabilities on $C_{0}^{n}(0,1)$ (the space of continuous $\mathbb R^{n}$ valued process on $[0,1]$ starting from zero) induced by two ...
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0answers
57 views

Rate of Convergence of Compound Poisson Laws to Infinitely Divisible Laws

It is known that every infinitely divisible random variable is the limit in law of a sequence of compound Poisson random variables (see for instance Theorem 1.2.18 of Lévy Processes and Stochastic ...
4
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1answer
116 views

Orlicz Norm and A result on expectation

I am reading paper which is mainly about Dobrushin's contraction coefficient and its generalization. In page 27, the following is defined: Consider arbitrary, non-negative, convex function ...
4
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1answer
171 views

Is the space of Radon measures a Prohorov space?

Consider the spaces $C_c(\mathbb{R})$ of compactly supported continuous functions equipped with the inductive limit topology and the Banach space $C_0(\mathbb{R}) = \overline{C_c(\mathbb{R})}^{\, ...
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0answers
78 views

Occupancy problem with limited capacity and two types of balls [closed]

I am considering the following problem that I suspect to be standard. One has a set of $N$ balls composed of a fraction $\alpha$ of red balls and $(1-\alpha)$ of black balls (we assume $\alpha N$ is ...
7
votes
1answer
150 views

Defining functions pointwise vs. almost everywhere (w.r.t. uncountably many mutually singular measures)

My question is motivated by a general measure-theoretic problem that one frequently encounters in probability: the need to work with uncountably many mutually singular measures at once, and with ...
2
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0answers
342 views

Homemorphism between $X$ and $\mathcal{P}(\mathcal{P}(X))$

Let $X$ be a topological space, $\mathcal{P}(X)$ be the set of all Borel probability measures on $X$. Endow the latter with the weak* topology. I was wondering whether there exists a (nontrivial) ...
2
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1answer
74 views

distance to median in terms of $L_1$ variance

Suppose that $X$ is a random variable with finite first moment and median $m$. Let $X'$ be an independent copy of $X$. What inequalities relate $E|X-X'|$ and $E|X-m|$? What is the best lower bound on ...
9
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2answers
289 views

An inequality for copulas

Suppose that $f$ from $[0,\infty]$ onto $[0,1]$ is completely monotonic on $(0,\infty)$, and let $g$ be the inverse of $f$. For $(u,v)$ in $[0,1]^{2}$, define $C(u,v) = f(g(u)+g(v))$, and let $a = ...
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0answers
31 views

Is it possible to distinguish between to edge orientation while learning a network structure?

I'm considering the case of learning bayesian network structure using a dataset $\mathcal{D}$ with scoring methods : $$\mathcal{G}^*=\max_{\mathcal{G}}\text{Score}(\mathcal{G}, \mathcal{D})$$ I'm ...
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0answers
84 views

On the decay of correlations of an ergodic sequence over the set $X_{0}=0$

The following question arose while I was trying to explore possible further extensions of a CLT by Liverani which I mentioned here already (see this link, I can tell you more details upon request). It ...
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0answers
22 views

assumptions on local rademacher complexities

A lot of the work on Local Rademacher complexities of Koltchinskii, and Bartlett for fast rates of convergence is based on Bousquet's version of Talagrand's inequality [1] (Theorem 2.11). However the ...
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3answers
308 views

Reference request: a guide through quantum probability

Could you point out a comprehensive reference book (or more than one, if it is the case) on Quantum Probability that introduces the subject and then gradually builds up to the edges of contemporary ...
3
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1answer
109 views

Lower bound for the $p$-th absolute moment of a sum of random variables

Suppose that $X_1,\ldots,X_n$ are independent random variables with $\operatorname E X_k=0$ and $\operatorname E |X_k|^p<\infty$ with $1<p<2$ for each $1\le k\le n$. I am interested in the ...
0
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3answers
135 views

An inequality based on expectation of continuous random variables

I am trying to prove the following statement: $$ E[g(X)] E[X^2g(X)]\ge E[Xg(X)] E[Xg(X)] $$ where $X$ is a random variable, $E[\cdot]$ denotes the expectation operator with respect to ...
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0answers
29 views

Reference request for specific POMDP examples

Following is strictly for discrete-time discrete-space Markov chain. Consider a partially observed Markov decision process (POMDP) $P = \{X,O,A,P,B_a\}$. Here $X = \{x_1, \cdots, x_n\}$ refers to ...
5
votes
3answers
116 views

Random partitions with prescribed pairwise membership probabilities

Let $(p_{ij}) \in [0,1]^{n \times n}$ be a given symmetric matrix, with $1$ on the diagonal. Suppose $\pi$ is a partition of $[n]=\{1,\dots,n\}$ and let us write $i \stackrel{\pi}{\sim} j$ if $i$ and ...
3
votes
1answer
209 views

Convergence of random variables with hypergeometric distribution

This is a very interesting conjecture of large scale property of hypergeometric distribution. Let $a>1$ be a integer constant, $N\in\mathbb{N_+}$, for any $x<N-1$, consider $N+(a-1)x$ balls in ...
5
votes
0answers
99 views

What's the variance in the Six Degrees model?

Recall the six degrees of Kevin Bacon game. You can even play the game at The Oracle of Bacon, and their search works via Breadth First Search. I interpret the punchline as saying that if I start ...
0
votes
1answer
98 views

Convergence in the Wasserstein metric and the square root function

Let $f$ be a smooth probability distribution on the unit square $S$ such that $f(x)>0$ on $S$. Let $\{g_i\}$ be a sequence of smooth probability distributions such that $g_i(x)>0$ on $S$ as ...