Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
0
votes
0answers
68 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
votes
1answer
88 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
votes
4answers
309 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 ...
1
vote
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
votes
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
votes
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 ...
-1
votes
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
votes
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 ...
0
votes
0answers
34 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 ...
1
vote
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
votes
1answer
160 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 ...
1
vote
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
votes
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
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
0answers
77 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 $0$:
$$\lim_{t\rightarrow 0} \mathbb E f(S_t) = f(0), \text{ if } \mathbb ...
0
votes
0answers
53 views
Derivative of the Expectation of an Integral over a Diffusion
I am trying to prove the following, where $S_t$ is a diffusion:
$$ \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) $$
Proof attempt:
Lusin's ...
0
votes
0answers
48 views
Consistency Conditions of the Kolmogorov Extension Theorem
Kolmogorov's extension theorem allows for the construction of a variety of measures on infinite-dimensional spaces, and its conditions are supposedly "trivially satisfied by any stochastic process". ...
5
votes
1answer
123 views
Asymptotic behavior of $X_n$ in a Dirichlet vector $(X_1, …, X_n)$
Let $(\alpha_k)$ be a sequence of positive numbers and let $(Y_k)$ be a sequence of independent random variables $Y_k \sim \text{Gamma}(\alpha_k,1)$. Set $X_n=\dfrac{Y_n}{\sum_{i=1}^nY_i}$.
(edit) ...
3
votes
2answers
189 views
Picking codewords that are close
I posted this question in http://math.stackexchange.com/questions/1142698/picking-codewords-that-are-close a week back.
Let $[n,k,d]$ be a linear code over $\Bbb F_q$ with minimum distance $d$ and ...
2
votes
0answers
104 views
Inequality with CDF of order statistics
here is a problem I have been struggling with for a while now. This is for a paper I am working on. Any help would be appreciated! Here we go:
Each bidder's valuation $\theta _{i},$ $i=1,...,N$, is ...
2
votes
1answer
111 views
Does $\int \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \to \Phi(u)$ imply that $f_t \to \delta_1$?
I'm looking at a family $(f_t)$ of densities of some continuous random variables and know that
$$\int_{-\infty}^{\infty} \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \xrightarrow{t \to ...
0
votes
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
301 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
votes
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
votes
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 ...
1
vote
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
votes
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
votes
2answers
233 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
votes
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 ...
5
votes
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
votes
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
votes
1answer
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})}^{\, ...
1
vote
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
votes
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
votes
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
votes
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 = ...
1
vote
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 ...
3
votes
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 ...
1
vote
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 ...
3
votes
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 ...
3
votes
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
votes
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 ...
0
votes
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 ...
