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)

2
votes
1answer
58 views

Estimating mean and variance of a distribution based on error-prone estimates of its cdf

Suppose I have some random variable $X$ taking values in $[a, b]$ with unknown distribution (I am happy to assume the distribution is smooth, though it would be nice to not have to). I have a ...
3
votes
0answers
91 views

Worst-Case Solution to (Stochastic) Matrix Inequality

EDIT: Some specific conjectures added. This problem comes with an associated stochastic process, but I phrase everything as linear algebra in case somebody from a non-probability community has seen ...
1
vote
0answers
44 views

Reference for a special case of the Hanson-Wright inequality

I would like find tail bounds for the expression $$ \begin{align*} \left|\left\langle a,\phi\right\rangle \left\langle \phi,b\right\rangle -\left\langle a,b\right\rangle\right|, \end{align*} $$ where ...
8
votes
3answers
290 views

Reference for a strong intermediate value theorem for measures

Let $\mu$ be a finite nonatomic measure on a measurable space $(X,\Sigma)$, and for simplicity assume that $\mu(X) = 1$. There is a well-known "intermediate value theorem" of Sierpiński that states ...
5
votes
1answer
318 views

lower-bound for $Pr[X\geq EX]$

Given n random variables, $X_1, ..., X_n$, each takes value 0 or $a_i \in[0, 1]$. $X = \sum_{i=1}^n X_i$ and $EX \geq 1$ is the expected value of $X$. Can we get a lower-bound for $Pr[X \geq EX]$? It ...
3
votes
1answer
55 views

Density for Translated Process

Let $M$ be a (compact) Riemannian manifold. Let $v$ be a smooth vector field on $M$ with flow $\Theta_t$. Let $L$ be an elliptic second order differential operator on $M$ that generates the Ito ...
0
votes
0answers
65 views

Approximation of quadratic variation

Here $M$ and $N$ are two bounded continuous martingales with respect to some filtration $(\mathcal F_t)_t$. I found this claim in a paper I was reading: $t$ being fixed, then a.s. $$\lim_{h\rightarrow ...
0
votes
0answers
30 views

Probability of close approach for multivariate normal variables

The following problem comes from a physical model of two groups of particles in three dimensions. I need to know the probability that the two groups of particles approach each other within some ...
1
vote
0answers
55 views

Asymptotic variance for partial sum of a stationary process

Let $X = (X_1, \dots, X_n, \dots)$ be a sequence of random variables. We assume that the process X is stationary i.e. for any integer $k$, any set of indices $i_1 < \dots < i_k$ and any integer ...
1
vote
0answers
59 views

Measurability of solution of diffusion equation in sub sigma algebra

I want to solve the following problem: Get $\omega \in \Omega \subset \mathbb{R}$, $x \in D \subset \mathbb{R}^2$ and $0<a_i\leq a(.,.)\leq a_x<\infty$. Let $a( x;. )$ and $f(x;.)$ be ...
3
votes
0answers
127 views

Donsker's Theorem for triangular arrays

I should mention that I already posed this question on Math Stack Exchange, but didn't receive much feedback. Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given ...
2
votes
1answer
101 views

Restricted singular values of random matrix

Let $X \in \mathbb{R}^{p\times p}$ be a large square matrix, consisting of i.i.d. Gaussian entries. Then it is known that the singular values of $X$ follow the Marchenko-Pastur law. Now let's ...
1
vote
0answers
88 views

First passage time of a pure drift process

I am facing the following unusual problem: $Z_t$ is a pure drift process of the form $$ dZ_t = \kappa(X_t - Z_t) dt $$ where $X_t$ is another bounded process. I am interested in computing / ...
5
votes
1answer
295 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 ...
0
votes
1answer
251 views

About an integral equation

I would like to obtain $g$ by solving the following integral equation $$ \int_s^T R(u) dg(u) + f(s,T)\int_s^T g(u)du =0$$ where $f,R:\mathbb R _+ ^*\rightarrow \mathbb R _+ $and $g: \mathbb R _+ ...
4
votes
1answer
205 views

Isomorphisms between spaces of test functions and sequence spaces

I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest ...
5
votes
2answers
304 views

Does this equation has a closed-form solution for $t$? ($(1-p)\sum_{i=0}^{n}t^i = p\sum_{i=0}^{n}(1-t)^i)$)

We are given $n\in \mathbb N^+$ and $p\in[\frac{1}{2},\frac{n+1}{n+2}]$. Our goal is to find $t\in[0,1]$ such that $$(1-p)\sum_{i=0}^{n}t^i = p\sum_{i=0}^{n}(1-t)^i$$ Is there a closed-form ...
0
votes
2answers
103 views

Probability spaces involved in using Bayesian Inference

I am currently reading "Statistical and Inductive Inference by Minimum Message Length" by C.S. Wallace. In this, Wallace gives a fairly informal account of Bayesian Inference which, in the case ...
3
votes
0answers
87 views

Ask for reference of a stochastic process

I would like to know whether the following stochastic process is well studied. Let $\{U_k: k \ge 1\}$ be a sequence of i.i.d random variable. $U_1$ is uniformly distributed on the unit interval $[0, ...
-3
votes
3answers
292 views

Determinant of matrix from set {-1, 1} [closed]

Let $A \in \mathbb{R}^{11 \times 11}$ and it's elements are form set $\{ -1,1 \}$. $\mathbb{P}(-1) = \mathbb{P}(1) = 0.5$. What is a probability to get such a matrix, that $\det A > 4000$? I have ...
6
votes
0answers
145 views

Is an infinite-dimensional “Lebesgue measure” uniquely determined by a set of positive finite measure?

Let $\mu$ be a probability measure on a subset $C \subset \mathbb{R}^\infty$ of the space of sequences, and assume, for simplicity, that $C$ is closed and convex. We say that $\mu$ admits shifts if ...
1
vote
1answer
132 views

How does Azuma's Inequality result from Pinelis Inequality?

According to [1] Let $(\mathcal{X},||\cdot||)$ be a separable Banach space and let $S(\mathcal{X})$ denote the class of all sequences $f=(f_j)=(f_0,f_1,...)$ of Bochner-integrable random ...
1
vote
0answers
42 views

Finding a general form of the density function when we have a four dimensional random variable

Consider a subject having time of the specific event $T_i$, which is a single sample from a distribution $F_i$ with density $f_i$ and support $[t_{\min},t_{\max}]$, for $i= 1,\ldots,n$. Let these ...
1
vote
2answers
133 views

Speed and absence of non-constant bounded harmonic functions

For a (symmetric) random walks on countable groups generated by $\mu$, there is a "brute-force computation" argument of Avez (1974) that shows that if the entropy $h_\mu$ is trivial then there are no ...
0
votes
0answers
94 views

Best possible concentration inequality in high dimensions

Let $X_1,\ldots,X_n$ be independent random variables in $\mathbb{R}^d$ with $EX_i=0$ and $||X_i||_{2}\leq 1$. What is the best known exponential upper bound for $$P(||X_1+\cdots+X_n||_{2}>x)?$$ In ...
2
votes
0answers
72 views

McDiarmid's inequality on normed spaces

McDiarmid's inequality says if a function $f: \mathcal{X}^n\to\mathbb{R}$ has the property that $$ \sup_{x_1,\dotsc,x_n,x'_i\in\mathcal{X}} ...
3
votes
1answer
115 views

Three-dimensional Apollonian spirals

Given mutually (externally) tangent spheres $S_1$, $S_2$, $S_3$, $S_4$, let $S_n$ be the unique sphere externally tangent to $S_{n-1}$, $S_{n-2}$, $S_{n-3}$, and $S_{n-4}$ for $n \geq 5$. Let ...
5
votes
0answers
154 views

A note on Doob's theorem

I have faced the following problem, regarding to the Martingale Theory. Because this area far from my area I don't know whether this problem is in literature or this can be simple question for ...
5
votes
1answer
374 views

Harmonic Crystal using Random Walk

Me and my advisor were looking for a specific proof of the disorder in $2d$ harmonic crystals. We could not find a paper or a textbook with it, so I thought trying my luck here. Basically, it is a ...
0
votes
1answer
63 views

Rademacher complexity of a Lipschitz class: Are the boundedness constraints necessary?

Consider the following function class: $F={f:R^d\rightarrow [a,b], f(x)=\sigma(w^Tx)}$ where $\sigma(.)$ is Lipschitz, and $w\in R^d$ is a parameter vector. The problem I'm working on is a machine ...
1
vote
1answer
79 views

Perturbation of a Bessel process of dimension 2

Bessel process of dimension 2 is defined to be solution of $$ dX_t=dB_t+\frac{1}{2X_t}dt,\quad X_0=x_0>0 $$ where $B$ is a standard 1-dimensional Brownian motion. $X$ can be viewed as the norm of a ...
8
votes
1answer
235 views

Algorithm to produce random number with a gamma distribution

I'd like to produce pseudo-random numbers with different distributions for a Monte Carlo simulation. I've got the poisson distribution working nicely with an algorithm from Knuth. I'm having trouble ...
8
votes
0answers
118 views

Randomly placing nonoverlapping unit cuboids

Suppose one places unit cuboids of dimension $d$ with min-corners uniformly distributed to lie in $[0,n]^d$, but with cuboid (strict) overlap forbidden. At some point, the region is "saturated," ...
2
votes
1answer
165 views

Random walks with exponential decreasing steps

Let $g$ be the golden number (or another algebraic integer in $(0,1)$ that fullfills an equation with coefficients $\pm 1$). Consider the random walk on $\mathbb{R}$ starting with $0$ and walking ...
5
votes
2answers
295 views

A central limit theorem for a trigonometric series involving primes

In some recent work I found I needed to prove a central limit theorem for the interesting series: $\sum_{n=1}^\infty \cos (u \log p_n) $ where u is a random variable uniform on the interval ...
1
vote
0answers
44 views

Is there a unique tilted measure with specified marginals?

Suppose $\mathcal{A},\mathcal{B}$ are finite sets and $\mu_{A,B}(a,b)$ is a probability measure on the product set $\mathcal{A}\times \mathcal{B}$ so that $\mu_{A,B}(a,b)>0$ for each $a\in ...
4
votes
1answer
93 views

Earth mover/Wasserstein distance between a pdf and an empirical distribution

This question is inspired by this much older question: Convergence of an empirical distribution w.r.t. the Hellinger distance Let $P$ be a continuous probability distribution on a compact subset of ...
0
votes
2answers
160 views

“Convolution” for Multiplying Random Variables

The following situation arises frequently in probability. Suppose we have two independent continuous random variables $X$ and $Y$ and we consider their sum, $Z=X+Y$. Then the pdf of $Z$ is the ...
2
votes
0answers
57 views

Controlling fluctuations in a Markov chain

For $N>0$, consider the Markov chain $x_n$ on $\{0,1/N,...,1\}$ that moves up by $1/N$ at rate $(c-x_n/2)N$ and down by $1/N$ at rate $(c+x_n/2)N$. As $N\rightarrow\infty$ sample paths approach ...
0
votes
1answer
65 views

Rank of a sequence of covariance matrices

Let $X_i$ ($i=1, \dots$) be an orthonormal basis for $L^2(\Omega, \mathbb P)$. In particular, it holds that $$\mathbb E[X_iX_j] = \delta_{ij}.$$ Now take $Z\in L^2(\Omega, \mathbb P)$ and define ...
0
votes
0answers
22 views

Literature on the notion of combining two discrete stationary processes with the latter process slowed down

Is there any literature about the following way of combining two stationary processes? Let $X_1, X_2, \dots$ be a discrete-time stationary process. Let $A$ be a subset in its sample space. Let ...
1
vote
1answer
112 views

Hoeffding's inequality for vector valued random variables

Is there a version of Hoeffding's inequality for vector valued random variables? This seems to be hard to find and I wonder why. I suppose it is difficult to show Hoeffding's lemma, since the proof ...
1
vote
1answer
84 views

Mixing time of a continuous time Markov chain with arbitrary rate matrix

I would like to calculate the mixing time of a continuous time starting from the rate matrix and not necessarily assuming that the time in between jumps have rate 1 - all I have is the (finite ...
1
vote
1answer
138 views

Probability of each edge in K-clique [closed]

For $c \in R$ and $k \in N$, $k \geq 3$ let $p_{k,c} := n^{\frac{−2}{k+1}}log^c(n)$. I would like to prove that exists $c\in R$ such that every edge in the random graph $G(n,p_{k,c})$ lies in a copy ...
5
votes
2answers
230 views

Regularity of random Fourier series

The following two statements appear to be true (but do correct me if I am wrong): The coefficients of a $C^k$ function on the torus $T^n$ decay at least as fast as $x^{-k}$ (where $x$ is some norm ...
3
votes
0answers
61 views

Probability that a random projection doesn't reduce the distance of a point from a subspace too much

Consider the natural uniform measure (is it called the Haar measure?) on the set of $(n-k)$-dimensional subspaces of $R^n$. We are given a $d$-dimensional affine subspace $U$ (think of $d, k \ll n$; ...
7
votes
5answers
452 views

Bound on sum of complex summands involving binomial coefficients

I am trying to find the asymptotic behavior of the sum: $$ \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^i y^{2n-i}$$ as $n\rightarrow\infty$. Here $x$, $y$ are complex numbers and I have ...
1
vote
1answer
208 views

Probability of connected graph on torus

Let $G = (V, E)$ be a graph on n vertices constructed in the following way: Each vertex $v \in V$ is positioned uniform randomly in $[0, 1] × [0, 1]$. Connect two vertices $u, v \in V$ if $d(v,u) ≤ ...
3
votes
1answer
109 views

Estimate for Levy metric

In the Encyclopedia of Mathematics there is an inequality for Levy metric ($d_L$): $$d_L(E,F) \leq \{\beta_r(F)\}^{r/(r+1)},$$ where $E$ is a a distribution that is degenerate at zero, $\beta_r(F)$, ...
2
votes
0answers
53 views

Fixing (non)-independency of a the subfamilies of finitely many events.

I'm would be interesting in any construction of a probability space with n events (n is given), where for every subset of these events, it is also given whether or not, the family is mutually ...