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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Fixed area, largest mass — is there a name?

Let $x\in \mathbb{R}^n$ and let $s_k(x)$ denote the sum of the $k$ largest entries of $x$. The function $s_k(x)$ is well-known to be convex and is often used in optimization, such as ...
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46 views

Advanced, pure and applied [on hold]

A clown is riding a single wheel cycle along a highwire from point A to point B. These two points are the same height, however as the clown cycles the highwire decreases in height to a minimum point ...
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23 views

How to put together a set of modified conditional distributions to a joint distribution? [on hold]

I am abstracting my original problem to a simple scenario.Consider a bi-variate multi-modal mixture of gaussian distribution, P(x,y). When we slice through x or y we get a univariate multi-modal ...
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0answers
14 views

Finding the bound of a mixture model percentile [on hold]

I could do with some help on the following issue. I'm trying to obtain $\alpha$ from: $\beta = \int_{\alpha}^{\infty} \sum_{i=1}^{n} p_i f_i(x) dx$ I have that: $0 \leq p_i \leq 1$, $\sum_{i=1}^{n} ...
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2answers
154 views

A balls and urns model for a hashing problem

Fix $N \in \mathbb{N}$. Suppose we throw $N$ numbered balls into $N$ numbered urns, so that for each $b \in \{1,\ldots,N\}$, ball $b$ lands in urn $j$ with equal probability $1/N$. Choose a number $c ...
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1answer
40 views

Multinomial proxy variables: Bound on probability of their sums

Suppose $(X_1,X_2,..X_i,..,X_b)$ as multinomial vector of random variables with $N=\sum_{i=1}^b X_i$ and probabilities $p_i$ to parametrize the $X_i$. Let us take the following imagination to ...
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45 views

Can the GUE be thought of as a uniform point in a high-dimensional polytope

I have thought about this question for a long time and could only find partial answers. The Gaussian Unitary Ensemble (or GUE) is the eigenvalues of a random Hermitian matrix with complex Gaussian ...
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77 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 $\{ ...
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1answer
89 views

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

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 ...
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4answers
327 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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84 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 ...
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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 ...
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56 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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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 + ...
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2answers
164 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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37 views

Finding a random variable with a density function [closed]

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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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 ...
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1answer
165 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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39 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 ...
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118 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 ...
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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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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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41 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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78 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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53 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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51 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". ...
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126 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) ...
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191 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 ...
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116 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 ...
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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 ...
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1answer
49 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. ...
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309 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 ...
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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 ...
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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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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 ...
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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 ...
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2answers
236 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 ...
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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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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 ...
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117 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 ...
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172 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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79 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 ...
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1answer
153 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 ...
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346 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) ...
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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 ...
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291 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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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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85 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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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
311 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 ...