Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

learn more… | top users | synonyms (1)

0
votes
0answers
33 views

Strong Markov vector-valued process from component strong Markov process and independence

I want to prove that if $X$ and $Y$ are (continuous time) independent strong markov $\mathbb{R}$-valued processes w.r.t. their natural filtrations $\mathcal{F}^X_t$ and $\mathcal{F}^Y_t$, that the ...
3
votes
1answer
134 views

Moment matching on the standard simplex

Let $\vec{\mu}_1, \vec{\mu}_2,\ldots, \vec{\mu}_k \in \Delta^{d-1}$ be $k\ (k\geq 2)$ distinct vectors on the standard simplex, where $$\Delta^{d-1} = \{\vec{\mu}\in R^{d}:\| \vec{\mu}\|_1 = 1,\mu_j ...
1
vote
0answers
46 views

Expected number of forward jumps to reach a given quantile of a rv [closed]

I'm a noob in randomized algorithm and ran into a problem(definitely not home work. I'm doing a self study out of my interest with help of my friends. I'm pursuing research career in a machine ...
3
votes
1answer
105 views

Learn a distribution from distributions on samples

There's many good ways to learn a distribution $p_X$ of an r.v. $X$ over $k$ symbols given many i.i.d. samples $X_1,\ldots, X_n$. The simplest is to use the sample relative frequencies $\hat{f}_X$ as ...
1
vote
0answers
46 views

Stochastic Ordering of Negative Binomial-like Distributions

Please forgive me if this is not precise enough to post here. Simply ask me to remove it if it is not suitable. I am new here. I am bounding the running time of an algorithm as a random variable $X$ ...
0
votes
0answers
88 views

Bounds on Wasserstein (Kantorovich) distance

Let $X$ be a Polish space endowed with a bounded metric $\rho_X$. Let $\mu, \mu'$ be two probability measures, and $\kappa, \kappa'$ be two stochastic kernels on $X$. Assume that $\kappa, \kappa'$ are ...
2
votes
0answers
677 views

What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)

In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required. In random signal theory, this distribution is typically a ...
3
votes
0answers
78 views

How does Jensen Shannon divergence and KL divergence correlate?

I am wondering if there is way to derive the correlation between Jensen Shannon divergence and KL divergence for two distributions: P and Q, in order to show that if JSD(P,Q) decreases, KLD(P,Q) ...
0
votes
0answers
67 views

Refined versions of Azuma's inequality

Is there any version of Azuma's inequality where the bound $c_k$ as mentioned in https://en.wikipedia.org/wiki/Azuma's_inequality comes in the numerator of the fraction in the negative expoential.
2
votes
1answer
96 views

An Inequality Regarding the Squared Conditional Variance

Given absolutely continuous random variables $(X, Y)$ with joint distribution $P_{XY}$, we construct $Z:=\sqrt{\gamma} Y+N_\mathsf{G}$ where $N_\mathsf{G}\sim N(0, 1)$ and is independent of $(X,Y)$. ...
6
votes
0answers
96 views

Extreme unitary minimal models of conformal field theory

Some of the best understood conformal field theories are the 2D unitary minimal models $\mathcal{M}(m+1,m)$ indexed by the integer $m\ge 2$ and with central charge $$ c=1-\frac{6}{m(m+1)}\ . $$ I ...
2
votes
0answers
71 views

Factors between IID on trees: what about the useless information?

Let $p \in (0,1)$. Take $E$ to be the edge set of the trivalent tree $T$, and $G$ to be the automorphism group of $T$. Let $f$ be any $G$-equivariant map from the measure space $([0,1]^E, ...
8
votes
1answer
278 views

Berry-Esseen bound for martingale sequence with varying and dependent variances

Let $(X_{1},\ldots,X_{k},\ldots)$ be a martingale difference sequence, i.e. $$ E[X_{k}|\mathcal{F}_{k-1}] = 0 $$ where $\mathcal{F}_{k-1}$ is the $\sigma$-algebra filtration at $k-1$. Let ...
2
votes
1answer
99 views

Random Walk 2D with dependent weights [closed]

I have spent a lot of time trying to solve this problem but have had no luck so far! Any help would be highly appreciated! Suppose I have a 3x3 grid as shown below. (3,1) (3,2) (3,3) (2,1) (2,2) ...
1
vote
0answers
73 views

Increase mutual information for binary symmetric channel

I have a question about increasing mutual information for the binary channel. Assuming there is an independently $K$ dimensional binary source signal denoted by $X=[X_1, X_2, \cdots, X_K]$, a parallel ...
0
votes
2answers
112 views

Different inner products for vector spaces of random variables

The inner product that appears in most books on probability is the covariance $\langle X,Y \rangle = E[XY]$ (considering that $X$ and $Y$ are zero mean real random variables). Are there other inner ...
1
vote
0answers
113 views

Interchanging integrals and continuous linear forms in RKHS

I am reading Reproducing kernel Hilbert spaces in probability and statistics by A Berlinet, C Thomas-Agnan. In Chapter 5 INTEGRATION OF $\mathcal{H}$-VALUED RANDOM VARIABLES they write One of the ...
0
votes
0answers
67 views

Proof of variance in wishart distribution

I wanna to prove the variance of wishart distribution, first a brief description of wishart distribution, how can i proof it? I wrote a solution but the result is not correct, please help me to fix ...
1
vote
0answers
132 views

A weighted ergodic average

According to my simulations, it looks like the number of times that the $N$ first iterates $u_0$, $\ldots$, $u_{N-1}$ of the sequence $(u_n)$ defined here meets an interval $I$ is close to $N|I|$ ...
3
votes
1answer
95 views

expected value of multiplication of matrices

I start with background and then ask my question, background is a brief description of wishart distribution. Background The Wishart distribution with $\nu$ degrees of freedom and positive definite ...
0
votes
0answers
55 views

Maximal inequality for Markov process

For a Markov process $\{X_n\}$ is there any inequality available for $$ E[\sup_{0 \leq n \leq k} X_{n}]$$ in terms of moments of $E[X_n], 0 \leq n \leq k$
1
vote
1answer
97 views

Do there exist random variables that force transitivity of dependence? [closed]

In general, statistical dependence is not transitive. If $Y$ and $X_{1}$ are dependent, and $Y$ and $X_{2}$ are dependent, then $X_{1}$ and $X_{2}$ are NOT necessarily dependent. However, in some ...
3
votes
1answer
88 views

How to show monotonocity and the limit? [closed]

Let me reformulate my recent question. Let $n, N$ denote density and cdf of Gaussian distribution. Let us consider its modification, given by density: $$\phi(x) = C\left\{ \begin{array}{lcc} ...
1
vote
1answer
139 views

Probability of covering a set

Suppose we have a set of $N$ numbers. At any given trial we can randomly choose $N^{1-a}$ of the numbers where $a\in(0,1)$. We replace the numbers back. How many trials does it take in average case ...
3
votes
1answer
167 views

What's the best betting strategy to double money if we have $\delta$ advantage?

Suppose that I am very skilled in a gambling game, and any day that I bet $x$, I get back $2x$ with probability $\frac 12+\delta$ (and nothing with probability $\frac 12-\delta$). My goal is to double ...
0
votes
0answers
22 views

Characterisation of non-Gaussian stationary stochastic processes via auto-correlation functions

It is well-known that a centred stationary Gaussian stochastic process is characterised up to equivalence by its autocorrelation function. Wiener, in his Time Series, makes the off-hand remark that ...
0
votes
0answers
26 views

Unidirectional continuous path discrete time random walk

Is there any material available to study on unidirectional continuous path discrete time random walk on a line interval. To say "unidirectional continuous path discrete time random walk on a line ...
2
votes
0answers
105 views

Infinite total variation of complex measure in Feynman path integral [closed]

I am trying to understand this: If one tries to define a Feynman path integral as a Wiener integral, then the complex measure could be of infinite total variation. What exactly does this mean? How ...
8
votes
11answers
1k views

What are fun elementary subjects in probability?

I have to read several lectures on probability or applications of probability for high school students (of high level). There is no necessary part I must lecture, that is, my aim is just ...
1
vote
0answers
65 views

Markov Chains and Simple Machine Learning [closed]

Suppose I have a large training set consisting of many strings of symbols. $TS = \{Str_0, Str_1, ..., Str_n\}$ $Str_i = \{Sym_0 ... Sym_{len}\}$ These strings of symbols are each generated by the ...
0
votes
0answers
57 views

CLT for sums of an infinite sequence of rv with an asymptotic distribution

Excuse me if the question is ill-posed. I'll do my best to explain the problem.I have a vector $(x^{(n)}_1, x^{(n)}_2, \ldots x^{(n)}_n),$ whose individual components can be shown to be asymptotically ...
1
vote
2answers
122 views

Probability of at most $K$ consecutive zeroes in a sequence of 0s and 1s [closed]

I want to prove that in a sequence W of length n, consisting of 1s and 0s, $P$( in $W$ there is at most $\frac{\log_2n}2$ consecutive zeroes ) $\leq \frac{K}{n} $ for some constant K. Can anyone ...
3
votes
1answer
67 views

Converging to moments obeying Carleman's condition

I believe that the following is true, and I'd like to make sure that it is and to have a reference. Suppose that $\mu_N$ are a sequence of measures on $\mathbb{R}$. Let $m_{N,k}$ be the $k$-th ...
1
vote
1answer
90 views

If the sample space is an Euclidean Space, we can use a different type of PDF

Reading this post, I realize that is possible to have another type of PDF (probability density function) in the special case when the sample space is an Euclidean space. Usually, we have a ...
1
vote
0answers
23 views

Properties of a map regarding the space of invariant probability measures for controlled Markov process

Let us consider a controlled Markov process with the transition kernel $p(dy|x,\theta)$ ($\theta$ being the control parameter. Now, consider the map $\theta \to I(\theta)$ where $I(\theta)$ is the ...
1
vote
0answers
54 views

How to solve the following bivariate recurrence?

$$F(n,r) = (1-w(r))F(n-1, r) + w(r-1)F(n-1, r-1)$$ where $w(r)$ is monotonically non-increasing in $r$ and $0 \leq w(r) \leq 1$ with $0 \leq r$ Initial condition: \begin{eqnarray} F(0, r) & = ...
4
votes
2answers
159 views

Does the truncated Hausdorff moment problem admit absolutely continuous solutions?

Let $\mu$ be a (Borel) probability measure on $[0,1]$ and define $m_j(\mu) = \int x^j\,\mu(dx)$. Let $k$ be a positive integer and consider the set $\mathcal C_{\mu,k}$ of probability measures $\nu$ ...
3
votes
1answer
101 views

Weighted global Holder property for Brownian motion paths

It is well-known that the Brownian motion (Wiener process) is almost sure locally $\alpha$-Holder for any $\alpha<1/2$. That is, with probability 1 $$ ...
4
votes
0answers
182 views

universality for large deviations?

This is a question about universality in probability theory, with combinatorics in mind. Consider a sequence of polynomials $P_n$ in one variable, with positive coefficients. Combinatorics is a large ...
2
votes
1answer
134 views

Brownian motion - probability of striking a sphere in $\mathbb{R}^n$ (a clarification)

This is primarily in reference to this question on MO. Serguei Popov's answer gives an explicit formula for the probability of a Brownian particle starting at the origin in $\mathbb{R}^n$ hitting the ...
1
vote
0answers
57 views

Sampling efficiently conditioned on linear constraints modulo both $\mathbb{F}_p$ and $\mathbb{F}_2$

Given a prime $p$ and positive integer $t \ll \log p$ (say $t = \sqrt{\log p}$), is there an algorithm that is polynomial time in $\log p$ to sample uniform $X, Y \in \mathbb{F}_p$ conditioned on the ...
7
votes
1answer
354 views

Does $|A+A|$ concentrate near its mean?

Fix $N$ to be a large prime. Let $A \subset \mathbb{Z}/N\mathbb{Z}$ be a random subset defined by $\mathbb{P}(a \in A) = p$, where $p = N^{-2/3 + \epsilon}$ for some fixed $\epsilon > 0$. My ...
5
votes
1answer
135 views

Stochastic Covering Number of a Convex Set

Consider a convex set, say $S = [0,1]^d$. Let $X_1, X_2,\ldots,X_n, \ldots$ be i.i.d. random variables that are uniformly distributed on $S$. Denote the Euclidean ball centered at $x \in \mathbb{R}^2$ ...
3
votes
1answer
181 views

How many times does a simple symmetric random walk of length n return to the origin?

Consider the simple symmetric random walk on the integers starting from the origin of length $n$. More precisely, I will denote an $n$ step random walk $w$ as $$ w:= \omega_0 \omega_1 \ldots ...
2
votes
0answers
69 views

Inverses of probability generating functions: positivity of derivatives

Let $\mathcal{G}$ be the set of probability generating functions of random variables taking positive integer values, considered as functions on $[0,1]$. So $G\in\mathcal{G}$ can be written ...
0
votes
0answers
66 views

Special random variables and monotone class theorem

I am currently reading a proof where the $\pi-\lambda$ Lemma and the monotone class theorem are applied to show a certain property for bounded random variables. The author of the book always shows the ...
4
votes
3answers
192 views

Measure of intersections in probability spaces

Let $(X,\mu)$ be a probability space, and $0<\epsilon<1/2$. Let $\{A_i:i\in \mathbb{N}\}$ be a collection of measurable subsets of $X$ such that $\mu(A_i)\geq \epsilon$ for all $i\in\mathbb{N}$. ...
0
votes
0answers
46 views

Maximal Correlation with Weak Gaussian Perturbation

Let a pair of random variables $(X,Y)$ be continuous random variables (i.e., they both have density with respect to Lebesgue measure) with joint distribution $P_{XY}$. The maximal correlation ...
0
votes
0answers
28 views

Hypergeometric distribution with a priori probabilities of the balls

If we have an urn with $N$ balls of two colours ($D$ red and $N-D$ black balls respectively), then the probability of having $k$ red out of $n$ balls drawn at once without replacement follows the ...
0
votes
3answers
173 views

Divergence of general random series and a special case

Is there any sufficient condition in terms of moments under which $$ \sum_{n=1}^{\infty} X_n$$ diverges a.s.?Here $X_n$ are not independent I am given that $\sum_n E[X_n]$ diverges. Actually, I am ...