# Questions tagged [pr.probability]

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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39 views

### Expectation of the exitpoint distance for the symmetric random walk

Let $\nu(x)$ be a symmetric probability measure with respect to the origin on $x\in[-1,1]$ such that $\nu(\{0\})\neq 1$.
Consider a random walk started at $S_0=0$, denoted $S_n=X_1+\dotsb+X_n$, ...

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31 views

### Optimizing a determinant

Any vector is assumed to be a column vector by default. Suppose $f(\mathbf{x})$ is the $d$-dimensional standard Gaussian density. I am interested in the following optimization problem:
$$
\max_g ~\...

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vote

**1**answer

48 views

### Expected determinant of random symmetric matrix with different Gaussian distributions of the diagonal and non-diagonal elements

Consider a random matrix $A \in \mathbb{R}^{N \times N}$ where the elements are random gaussian variables. The mean and variance of the elements are different on the diagonal and the off-diagonal:
$\...

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20 views

### Approximating Uniform/Arbitrary distribution with Gaussian mixture model?

so Silverman in his 1986 book mentioned about approximating distributions with Gaussian mixture models but he didn't go much further into the topic...I'm just wondering, say I'm given a N-dimensional ...

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votes

**1**answer

60 views

### How far away is $\max_{x: x \in \{0, \ldots, N\}} |W(x/N)|$ from $\max_{0 \leq t \leq 1} |W(t)|$ ($W(t)$ a Wiener process)?

How far away is
$$\max_{x: x \in \{0, \ldots, N\}} \left|W\left(\frac{x}{N}\right)\right|$$
from
$$\max_{0 \leq t \leq 1} |W(t)|$$
In other words, if you simulate a Wiener process over a finite ...

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37 views

### $P(\max_{0 \leq t \leq 1} \|W(t)\| \leq x)$ has no closed-form expression… right?

$P(\max_{0 \leq t \leq 1} \|W(t)\| \leq x)$ shows up in a formula for computing $p$-values for a certain statistic, where $W(t)$ is a $d$-dimensional (standard) Wiener process. My advisor says the ...

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**1**answer

53 views

### Maximum of sums of iid $X_i$'s where $X_i$ is the difference of two exponential r.v

Given $X_i = A_i - B_i$ where $A_i\sim \text{ Exp}(\alpha)$ and $B_i \sim \text{ Exp}(\lambda)$. Define $S_k = \sum_{i=1}^k X_i$ with $S_0 = 0$, and
$$M_n = \max_{1\leq k \leq n} S_k.$$
Is it ...

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**2**answers

108 views

### Continuous embedding of the Skorohod space D(0,1) into L^2(0,1)

Let $D(0,1)$ be the Skorohod space with the Skorohod topology, i.e. the space of real-valued càdlàg-functions on $[0,1]$ with topology induced by the metric
$$d(f,g) = \inf_{\varphi \in \Lambda} \left\...

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87 views

### Large deviation of random walk

1) Let $\{X_i\}_{i=1}^n$ be i.i.d. such that $\Pr(X_i=1 )=1-\Pr(X_i=-1)=p$. Define the random walk
$$
S_i = \sum_{j=1}^iX_j
$$
for $i=1,2,\ldots,n$.
I am looking for "good" exponential upper bounds ...

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votes

**1**answer

60 views

### Hoeffding's inequality for Hilbert space valued random elements

Suppose that $\mathbb H$ is a separable Hilbert space and $X_1,\ldots,X_n$ are independent zero mean $\mathbb H$-valued random elements such that $\|X_i\|\le s$ for each $1\le i\le n$, where $\|\cdot\|...

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**1**answer

93 views

### A problem related to the comparison of two integer-valued random variables

Consider an urn containing red, blue and green balls (the situation is the same illustrated in this post).
Let $X$ be the non-negative, integer-valued random variable defined as the number of trials (...

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**1**answer

180 views

### Complicated bound after using Stirling's approximation

I have this inequality $$\frac{1}{a}\exp\bigl\{-\frac{4}{h^2}\bigr\} \geq \frac{1}{f}$$ where $$ a \leq \Bigl(\pi^{d/2}\Gamma(\frac{1}{2}d+1)^{-1} + 1\Bigr) \left(\frac{h^{d+1}}{2} \Gamma \left(\frac{...

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**2**answers

112 views

### Example of measure for some algebra over N

$\mathcal F$ is set of events. Can you give an example of some algebra $\mathcal A$ over $\mathbb N$ and a non-zero finitely additive measure $\mu$ on this algebra, which has a countably additive ...

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34 views

### Extension of a result about measurable, additive functionals

Let $W$ be a set, and let $v$ be a finitely additive probability measure on $2^W$.
Equip $2^W$ with the Borel sigma-algebra $\mathcal{B}$ generated by the sub-basic sets of the form $\{a: w \in a\}$ ...

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**4**answers

199 views

### Apply doubly stochastic matrix M to a probability vector, then entropy increases?

Consider a vector $p =(p_1,...p_n)$, $p_i>0$, $\sum p_i = 1$
and a matrix $M_{ij}$, which is doubly stochastic: $\sum_i M_{ij} = 1, \sum_j M_{ij} = 1, M_{ij} > 0$.
Question 1 Just apply ...

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**0**answers

21 views

### How to formally relate the Parry Markov chain to a uniformly chosen bi-infinite path

I tried reformulating the comment after Proposition 10 in here as follows:
Let G be an aperiodic irreducible graph with adjacency matrix $A \in \mathcal{M}_d(\{0,1\})$ and associated Parry matrix $P$....

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39 views

### Matching Supply and Demand in an Uncertain World [closed]

i am trying to solve a question related to operations management when there is uncertainty. I am facing problem in determining the sample size and then what if the situation was not uncertain ? The ...

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**0**answers

43 views

### Concrete Hanson-Wright inequality?

I'm working on a paper that requires bounding
$$\Pr\left[|\vec x^\top Q \vec y| >= t\right]$$ where $Q$ is a matrix (happens to be symmetric) and $\vec x,\vec y$ are iid real mean-zero subgaussian ...

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**1**answer

81 views

### Measurability of C([0,1]) for the completion of the Wiener measure

Consider the completion $(\mathbb{R}^{[0,1]}, \mathcal{B}, \mu)$ of the Wiener measure on $\mathbb{R}^{[0,1]}$ (with the cylinder set $\sigma$-algebra).
Is the following true :
$C([0,1])\in \...

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102 views

### Expected distance of nearest matching pair in the game of pairs

Recently I was playing several rounds of the game of pairs with my children. I was surprised that almost every time, one matching pair was adjacent (either next to each other in a row, or vertically). ...

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**1**answer

105 views

### Shannon problem

Since a few days, I try in my research to model / formalize a source of Shannon a little weird, and I can't do it at all. First of all, I explain to you its operating principle and then I describe it ...

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125 views

### Statistical models of functions

I did a quick literature search and found nothing on "statistical models of functions". Let me explain what I am looking for. Given the category of Sets and Function, we have arbitrary functions ...

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**1**answer

90 views

### Expected values of two non-negative, integer-valued random variables related to an urn problem

Consider an urn containing $c$ distinguishable balls, $\alpha$ of which are red, $\beta$ of which are blue, and $\gamma$ of which are green, and $\alpha+\beta+\gamma=c$. We assume $\alpha,\beta,\gamma&...

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**1**answer

36 views

### Characterisation of a superset of the simplex

Does there exist a nice description of the following set:
\begin{equation}
A:=\left\lbrace x\in\mathbb{R}^{n}\ \colon\ 0< x_{i}-\bar{x}+\frac{1}{n}< 1\ \text{for} \ i=1,\dots,n\right\rbrace,
\...

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**1**answer

82 views

### Is a Gaussian measure on a Hilbert space determined by the coarser topology induced by the covariance operator?

I have a basic question about Gaussian measures on a Hilbert space:
Let $\mu$ be a non-degenerate Gaussian measure on a Hilbert space $(H_0,\left\langle \cdot,\cdot \right\rangle_0)$. Then the ...

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11 views

### Robust estimation of mean with total-variation uncertainty region

Let $\mathcal X$ be space and $P$ be a probability measure thereupon. Let $\mathcal B_{n,\delta}$ be the set of distributions $Q$ on $\mathcal X$ such that $D_f(Q\|\hat{P}_n) \le \delta$, where $D_f(\...

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62 views

### How can we treat the generator of a discrete-time Markov chain as the generator of a Markov-jump process?

In the popular paper Weak Convergence and Optimal Scaling of Random Walk Metropolis Algorithms by Roberts, Gelman and Gilks, the authors state (see below) that "in the Skorokhod topology, it does not ...

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**1**answer

99 views

### Is there a coupling that induces a given coupling via a transition kernel?

Let $X,Y$ be two measurable spaces, $\mu,\nu$ two probability measures on $X$, and $\kappa$ a transition kernel from $X$ to $Y$.
Define $\tilde\mu(dy)=\int_X\kappa(dy|x)\mu(dx)$ and $\tilde\nu(dy)=\...

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**1**answer

160 views

### Mode of a sum of Bernoulli random variables

Let $S_n=\tau_1+\cdots+\tau_n$ be a sum of independent Bernoulli random variables such that $\mathbb{P}(\tau_i=1)=p_i$. Is it true that the mode of $S_n$ is either its mean rounded up or rounded down?

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105 views

### Probability distribution from equidistribution - I

Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$. Denote $N_r(a,b)$ to be minimum $\ell_r$ norm of vector $(x,y)$ as $(x,y)$ ranges over all non-zero integral solutions to $(x,y)\equiv t(a,...

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79 views

### Probability density from standard domain

Pick $x+iy$ at random with respect to hyperbolic measure from $\{z:|z|\geq1,|\mathcal R(z)|\leq\frac12\}$. What does the probability distribution function of $\frac1{\sqrt y}$ look like?

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138 views

### Is the covariance of squares always bounded from below by two times the covariance?

I came across the following inequality in one of my calculations ($X,Y$ are centered random variables):
$$\operatorname{E}(X^2Y^2)-\operatorname{E}(X^2)\operatorname{E}(Y^2) \geq 2 \operatorname{E}(...

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**1**answer

46 views

### Generalization of Komlós–Major–Tusnády Approximation

The Komlós–Major–Tusnády Approximation (see Wikipedia) considers the sum of uniform variables in $(0,1)$. There are also version where instead the sum of equiprobable $0/1$ variables is used ($p=1/2$)....

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246 views

### Expectation of exponential of a function of independent Rademacher r.v.'s involving the error function

Let $Z,Z'\in\{-1,1\}^n$ be two independent vectors of i.i.d. Rademacher r.v.'s, where $1\leq n \leq d$ are two integers ($d\gg 1$). I am trying to get an upper bound on
$$
\mathbb{E}_{ZZ'}\left[ \exp\...

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32 views

### On the distribution of a random point of a poisson process

Let $T = \{t_i\}_{i=1}^\infty$ be the set of points in a Poisson point process on the positive half-axis with parameter $\lambda$, $I \in \mathbb{N}$ be a positive integral random variable with ...

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**1**answer

92 views

### Why the Fisher information matrix is equal to the Hessian matrix of the Kullback–Leibler distance at the true parameter?

I'm reading 《Algebraic geometry and statistical learning theory》.My problem is why the Fisher information matrix is equal to the Hessian matrix of the Kullback–Leibler distance at the true parameter?...

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67 views

### Exchangeable Bernoulli random variables with bounded summation implies negative correlation?

Let $\big\{X_1, X_2, ..., X_n \big\}$ be $n$ jointly exchangeable Bernoulli random variables, i.e., exchanging the order of these random variables does not change the joint distribution. If we know ...

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141 views

### Probability of one species reaching zero before the other in a Markov process on a 2d lattice

$\textbf{Background}$: Say we've got a two-variable system of stochastic chemical reactions, with quantities $\vec{x}(t) = (x_1(t),x_2(t)) \in \mathbb{N}^2$ evolving according to the following system, ...

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47 views

### Upper bound on expectation of product

I want to upper-bound the following quantity:
$$\mathbb{E}_Y\left[f(Y)g(Y)\right] $$
The idea would be to get something of the shape: $\mathbb{E}_Y[f(Y)]\cdot h(Y)$
where $h(Y)= j(\mathbb{E}_Y[k(g(Y))]...

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60 views

### Probability of the intersection of many events

Let $\{X_i\}_{i=1}^n$ be a sequence of $n$ i.i.d. Bernoulli random variables such that $\Pr\{X_i=1\}=1-\Pr\{X_i=0\}=p<1/2$.
I am interested in finding upper bounds on the probability
$$
\Pr\left\{\...

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27 views

### Gaussian Processes and the Measures of Finite Support Monad

Gaussian processes are defined on a domain, which is a set, and map each set element to a gaussian distribution over a target domain. I believe the idea is to model a function $f: X \rightarrow Y$ ...

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46 views

### Hitting times of Markov chains

I'm looking for references on the above topic. My particular interest is in discrete time, countable state space Markov processes. References can take any form (papers, books, notes etc.) and can ...

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72 views

### Tight bound on data binning problem

Consider dependent random variables $(X,Z)$ with joint pmf $p(x,z)$ on the finite alphabet set $\mathcal{X}\times\mathcal{Z}$.
Let $X^n, Z^n$ be $n$ i.i.d. repetitions of $(X, Z)$, i.e.,
$$p(x^n,z^n)=...

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107 views

### Which probability distribution has the most outliers?

Let $k$ be a positive real number. Which probability distribution over $\mathbb R$ maximizes $P(|x-E(x)|>k\cdot \operatorname{std}(x))$?

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106 views

### Distribution of the individual coordinates of a uniform random vector on a high-dimensional sphere

Let $X=(X_1,\ldots,X_n)$ be a random vector uniformly distributed on the $n$-dimensional sphere of radius $R > 0$. Intuitively, i think that for large $p$ every coordinate $X_i$ is normally ...

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32 views

### If $(Y_n)_{n\in\mathbb N_0}$ and $(N_t)_{t\ge0}$ are stochastic processes, what is the filtration generated by $\left(Y_{N_t}\right)_{t\ge0}$?

Let
$(\Omega,\mathcal A)$ and $(E,\mathcal E)$ be measurable spaces
$(Y_n)_{n\in\mathbb N_0}$ be a $(E,\mathcal E)$-valued stochastic process on $(\Omega,\mathcal A)$
$(N_t)_{t\ge0}$ be a $\mathbb ...

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**1**answer

259 views

### Maximal inequality for the average of i.i.d. random variables

Let $Z_i$ be i.i.d. random variables with $\mathbb{E}[Z_i] = 0$ and $\mathbb{E}|Z_i|^p< \infty$ for $p=1,2,3,\cdots$. I am looking for the following type of estimate if possible, and it is not like ...

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39 views

### Is there a transient graph whose spectral dimension two?

Let $G = (V(G), E(G))$ be an infinite connected simple graph.
Let $((S_n)_n, (P^x)_{x \in V(G)})$ be the simple random walk on $G$.
Let $p_n (x,y) = P^x (S_n = y)$.
A spectral dimension of $G$ is ...

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53 views

### Construction of Feller's pseudo-poisson process

Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(E,\mathcal E)$ be a measurable space
$(Y_n)_{n\in\mathbb N_0}$ be a $(E,\mathcal E)$-valued time-homogeneous Markov chain on $(\...

**0**

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**0**answers

79 views

### Supremum over all invariant Borel probability measures of the ergodic averages ratio of rates

Let $M$ a two-dimensional compact manifold and $f:M\to M$ a diffeomorphism $C^r$, $r\geq 2$ and $f(x,y)=(mx,\lambda y)$ where $m:M\to \mathbb{R}$ and $\lambda:M\to \mathbb{R}$ ,$\lambda<1<m$.
...