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.
8,632
questions
2
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0
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225
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An inequality regarding centered Bernoulli random variables
Let $\left\{V_1,\ldots,V_n\right\}$ be a set of (possibly dependent) identically distributed Bernoulli random variables, with
$$ p = \mathbb{P}\left(V_i=1\right) = 1-\mathbb{P}\left(V_i=0\right),\quad ...
3
votes
0
answers
76
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Using Kac-Rice formula to count average number of sub-regions carved out by $n$ random hyper-planes
Context. This is the first in a set of tiny pieces of a problem I've formulated to help me measure the "complexity" of certain piecewise linear functions. Thanks in advance for your help and patience.
...
8
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2
answers
1k
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Talagrand's inequality for the discrete cube
Talagrand showed that if $f$ is a convex $1$-Lipschitz function on $\mathbb{R}^n$, and if $\mu$ is a product of probability measures supported over the interval, then $f$ has Gaussian concentration w....
9
votes
1
answer
319
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Visibility in a growing orchard
This is a variant on Polya's orchard problem.1,2
Suppose trees are planted randomly in the plane.
The question is: How many trees are visible from the origin as
their radii grow?
More precisely, ...
1
vote
1
answer
221
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Barycenter Map on Wasserstein Space
Let $(X,d)$ be a complete separable metric space, $P_1(X,d)$ be the set of Radon probability measures on $X$ satisfying
$$
P_1(X,d)\triangleq \left\{
\nu:\,(\exists x_0\in X)\, \int_{x\in X} d(x,x_0)d\...
3
votes
2
answers
615
views
Lifting a probability measure to the power set
Let $X\neq\emptyset$ be a set and let $\mu:{\cal P}(X)\to [0,1]$ be a probability measure. Is there a probability measure $$\bar{\mu}:{\cal P}({\cal P}(X))\to [0,1]$$ with the following property?
...
7
votes
1
answer
483
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Understanding Gillman's proof of the Chernoff bound for expander graphs
My question is about the proof of Claim 1 in this paper: Gillman (1993).
At the end of the proof, the author says:
The matrix product $U^\top\sqrt{D^{-1}}(P+(\mathrm{e}^x-1)B(0)-\mu I)\sqrt{D}U$, ...
4
votes
1
answer
232
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Operator version of Birkhoff ergodic theorem
Suppose that $(\Omega,\mathcal{E},P)$ is a probability space and suppose that we have a measurable operator $T:\Omega\to\Omega$.
Recall that $T$ is said to be egodic if:
$T$ is measure preserving: ...
11
votes
2
answers
807
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Computing the sum of an infinite series as a variant of a geometric series
I came across the following series when computing the covariance of a transform of a bivariate Gaussian random vector via Hermite polynomials and Mehler's expansion:
$$
S = \sum_{n=1}^{\infty} \frac{\...
4
votes
1
answer
188
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Local central limit theorem far from the center
Let $X_i$ be a sequence of iid random variables, $E [X] = 0$, $E [X^2] = 1$ and $E [|X|^k] < \infty$ for some $k \ge 3$.
Classical local CLT says that the density function $f_n$ of $\frac1{\sqrt n}...
4
votes
2
answers
185
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Uniform convergence of averages for stationary ergodic process
Let $\{X_t, t\in\mathbb R\}$ be a well-behaved$^*$ stationary ergodic process.
I'm interested in the uniform convergence of averages:
$$
\sup_{|x|\le R_n} \left|\frac1{2n}\int_{x-n}^{x+n} X_t dt - \...
2
votes
1
answer
239
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Ratio of expectation involving random unit vectors
Let $u=(u_1,...,u_n), v=(v_1,...,v_n)$ be two random vectors independently and uniformly distributed on the unit sphere in $\mathbb{R}^n$. Define two other random variables $X=\sum_{i=1}^nu_i^2v_i^2$, ...
6
votes
2
answers
537
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An expansion from Ramanujan related to birthday problem
A friend designed a drinking game with a lucky wheel of 30 distinct icons. When playing, each one takes turn to spin the wheel, and write down the items until the first one who gets the item that has ...
6
votes
1
answer
1k
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Lipschitz function of independent subgaussian random variables
This question was asked here, but I have reason to believe that it's a serious research question appropriate for this forum (also, the answers given at the link aren't satisfactory).
If $X\in\mathbb{...
0
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0
answers
81
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A closed form of mean-field equations
Assume that a system at time t, for example number of costumers in a line at time $t$ which is denoted by $q(t)$, follows a Markov chain with these dynamics (probabilities)
$$P(q(t+\Delta t)-q(t)=1)=\...
4
votes
1
answer
1k
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Normal multivariate orthant probabilities
(Previously I posted a similar question on math.SE, hoping that this question would have an easy answer. As the question appears hard, I am hoping I can perhaps get more feedback here.)
Let $\mathbf{...
6
votes
0
answers
117
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What can be said about percolation clusters after deleting a positive fraction of edges in general?
Start with a bond-percolation process just above criticality, say $p=1/2+\varepsilon$ on the graph $\mathbb Z^2$ with $\varepsilon>0$.
Sample $D\in\{0,1\}^E$ from an independent product measure ...
2
votes
3
answers
951
views
Sum of Square of the Eigenvalues of Wishart Matrix
Let $A\in\mathbb{R}^{m\times d}$ matrix with iid standard normal entries, and $m\geqslant d$, and define $S=A^T A$.
I want to have a tight upper bound for $\sum_{k=1}^d \lambda_k^2$, where $\...
2
votes
0
answers
207
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Minimising an Integrated Relative Entropy Functional
Suppose I am given
A probability distribution on $\mathbf R^d$, with density $\pi (x)$.
A family of transition kernels $\{ q^0 (x \to \cdot) \}_{x \in \mathbf R^d}$ on $\mathbf R^d$, with densities $...
7
votes
1
answer
916
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Reference request: norm topology vs. probabilist's weak topology on measures
Let $(X,d)$ be a metric space and $\mathcal{M}(X)$ be the space of regular (e.g. Radon) measures on $X$. There are two standard topologies on $\mathcal{M}(X)$: The (probabilist's) weak topology and ...
0
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0
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55
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What is the probability that $X_i$ is the $k^{th}$ order statistic in consecutive trials?
Consider $n$ r.vs ${X_1, X_2,...,X_n}$. Each is i.i.d drawn from some distribution $f(.)$. What is the probability that $X_i$ is the $k^{th}$ order statistic in any two consecutive trials?
14
votes
1
answer
799
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Large-n limit of the distribution of the normalized sum of Cauchy random variables
What is the large-n limit of a distribution of the following sample statistic:$$x\equiv\displaystyle\frac{\sum^{n}X_{i}}{\,\sqrt{\,\sum^{n}X_{i}^{2}\,}\,}$$ when sampling the Cauchy(0,1) distribution? ...
4
votes
0
answers
198
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A lower bound on the expected sum of Bernoulli random variables given a constraint on its distribution
Given a set of Bernoulli random variables $x_1, \dots, x_n$ (not necessarily identical) with $X= \sum_{0<i\leq n} x_i$, I am intrested in finding a lower-bound for $\frac{\mathbb{E} [ \min (X,k) ]...
1
vote
0
answers
104
views
Hashed coupon collector
The story:
A sport card store manager has $r$ customers, that together wish to assemble a $n$-cards collection.
Every day, a random customer arrives and buys his favorite card (that is, each customer ...
2
votes
0
answers
108
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Arithmetic structure of non-zero cumulants
It is known that any non-Gaussian distribution must have infinitely many non-zero cumulants (Marcinkiewicz). I was wondering if something stronger is known about the structure of non-zero cumulants. ...
2
votes
0
answers
94
views
Distribution of a linear pure-birth process' integral
I stumbled across the following random variable, defined as the integral of a linear pure-birth process i.e. a Yule process:
$$
Z_t = \mathbb{E}\bigg[\int_0^t Y_s ds \bigg| Y_t=k , Y_0=1\bigg]
$$
...
1
vote
1
answer
259
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Symbol for monotone relationship between two probability distributions
Motivation:
At the present time it really isn't clear to me why this question might be inappropriate for the MathOverflow. However, it appears that some people are down-voting this question even if ...
3
votes
0
answers
92
views
Probability of a random collection of subsets being a cover
Consider the set $[n]=\{1,2,\ldots,n\}$. Suppose for each set $A\subseteq [n]$ I have a $p_A \in [0,1]$. I now create a random collection $\mathcal{W}\subseteq\mathcal{P}([n])$ of subsets of $[n]$ by ...
2
votes
0
answers
107
views
Average number of pieces of a random piecewise-linear function
Let $I$ be a (nonempty) compact interval in $\mathbb R$ and $a_1,b_1,\ldots,a_L,b_L \in \mathbb R$. Let $\varphi$ be a piecewise function with $T \ge 2$ pieces(for example $T=2$ for the choice $\...
3
votes
1
answer
2k
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Understanding Finite Size Scaling in Percolation Theory
Fundamental results in percolation theory are all based on the assumption that the system sizes are infinite, as the spanning/percolating cluster is by definition an infinitely sized cluster that ...
1
vote
0
answers
1k
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Linear Independence of random binary vectors
Suppose we have $Y_1, \ldots, Y_n \in \mathbb{R}^m$, $n$ independent random vectors ($m \geq n$), where the entries of each $Y_i$ are i.i.d. Bernoulli random variables taking the values $\{0, 1\}$ ...
7
votes
2
answers
258
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Slowest initial state for convergence of finite birth-and-death Markov chains
Consider the continuous-time birth-and-death Markov chain on $\{1,\cdots,n\}$ with all rates equal to $1$. Is it true that the convergence to equilibrium, in total variation distance, is slowest when ...
0
votes
3
answers
870
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Reference request: book on stochastic calculus (not finance)
I am looking at fractional Gaussian/Brownian noise from a signal theoretic and engineering point of view. In particular, I am looking at the math behind what defines these noise processes and what ...
10
votes
0
answers
367
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Concerning Luzin-(N)-property
Definition: a function $f:\mathbb{R}\to \mathbb{R}$ has Luzin-(N)-Property if $f$ maps any null set to a null set.
By https://www.encyclopediaofmath.org/index.php/Luzin-N-property, it is known that ...
5
votes
1
answer
166
views
Ratio of integrals with increasing dimension over Euclidean balls
Let $f_n(x)\geq0$ be any sequence of nonnegative $L^1(\mathbb{R}^n)$ functions such that $\int_{\mathbb{R}^{n}}f_n(x)dx=1$ where $dx$ is the Lebesgue measure on $\mathbb{R}^n$. For any $a>1,\...
3
votes
1
answer
611
views
Lower bound of the expectation of the product of inner products of random vectors
I encountered the following value in my research:
Let $n,m$ be some integer. Suppose $\alpha_1,\dots,\alpha_m$ are unit vectors in $\mathbb{R}^n$.
Denote
$$
L = \mathop{\mathrm{E}}_x[ \prod_{1\...
6
votes
1
answer
127
views
The distribution of the area of a region cut out by chordal SLE?
Let $\mathbb{D}$ be the unit disc. Let $a,b \in \partial \mathbb{D}$. Let $\gamma$ be a chordal $SLE_{k}$ from $a$ to $b$.
For $k \leq 4$, $\gamma$ is a simple curve, and so $\mathbb{D} \setminus \...
5
votes
1
answer
234
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The square modulus of coordinates of a uniformly chosen point in complex projective space is uniform in the simplex
I can't recall where I learned this (beautiful) fact, and I would like a reference (if possible, in a textbook):
Let $(z_0:\cdots:z_n) \in \mathbb{P}^n(\mathbb{C})$ be chosen uniformly at random w....
2
votes
0
answers
78
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Bridging between Rosethal Inequalities and log convex tails
Let $X_1,\ldots,X_n$ be independent with $\mathbf{E}[X_i] = 0$ and $\mathbf{E}[|X_i|^t] < \infty$ for some $t \ge 2$. Write $\|X\|_p = (E|X|^p)^{1/p}$.
Then we have the classical "Rosenthal-type ...
5
votes
0
answers
303
views
Points of continuity of Kullback-Leibler divergence with respect to weak convergence
I know that the Kullback-Leibler
$D(\mu||\nu) := - \int_K\log\big(\frac{d \nu}{d \mu}\big) \, d\mu,$
over probability measures on a compact $K$ subset of $\mathbb{R}^d$, is only weakly lower ...
6
votes
1
answer
546
views
Sub-Gaussian decay of convolution of $L^1$ function with Gaussian kernel
I think it might be helpful to put the new statement at the beginning and put the original post at the end. This new statement is more mathematically elegant.
Let $f\geq0$ be in $L^1(\mathbb{R}^d)$ ...
2
votes
0
answers
91
views
Defining weak solutions to infinitely many SDEs on the same probability space
Suppose I have an SDE of the form
$$dX_t=b(X_t)dt+\sigma (X_t)dB_t+\int_{\mathbb{R}}G_{t-}(y)N(dtdy)$$
which I can solve weakly if I cut off the last integral to range over the set $\{\mid{y}\mid > ...
5
votes
1
answer
170
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Reference request: When is the variance in the central limit theorem for Markov chains positive?
I'm looking for a reference which gives sufficient conditions for the variance to be positive in the central limit theorem for Markov chains (cf https://en.wikipedia.org/wiki/...
3
votes
1
answer
166
views
Maximal correlation and independence
Let $X$ and $Y$ be random variables. Then the maximal correlation $\rho_m(X;Y)$ is defined as
$$ \rho_m (X;Y) := \max_{(f(X),g(Y))\in S} \mathbb{E} [f(X)g(Y)] $$
where $S$ is the collection of pairs ...
4
votes
0
answers
131
views
For a martingale $f_0,f_1,\ldots $ how can we bound $P(\frac{1}{n} \|f_n\| \le 1$ for all $ n \ge N)$?
Suppose $f_0,f_1, \ldots$ is a martingale (or i.i.d sequence) in $\mathbb R^d$ with $f_0=0$ and all $\|f_n - f_{n-1}\| \le L$ say. There are many concentration results for the initial segment of the ...
7
votes
1
answer
779
views
Trace of inverse of random positive-definite matrix in high dimension?
Consider a random matrix $A \in \mathbb{R}^{n\times n}$ with i.i.d. entries, with symmetric law and finite variance. I am curious about the behavior of $$\mathrm{Tr}( (A^T A + \lambda \mathrm{Id})^{-1}...
3
votes
1
answer
184
views
Isoperimetry on $[0, 1]^n$ w.r.t $\ell_p$ distance, with $p \in [1,\infty]$
Let $A$ be a measurable subset of the metric space $\mathcal X = ([0, 1]^n,\ell_p)$ with $1 \le p \le \infty$, and define its $\varepsilon$-blowup by $A^\varepsilon:=\{x \in \mathcal X \mid \|x-a\|_p \...
6
votes
1
answer
198
views
Good upper-bound for $\mathbb E[|X-np|^r]$ where $X \sim \text{Binomial}(n,p)$ and $r \ge 1$
Disclaimer. Question moved from SE.
Setup
Let $X \sim \text{Binomial}(p, n)$, and $r \ge 1$.
Question
What is a good upper-bound for $\mathbb E[|X-np|^r]$ ?
Solution for small $r$
If $r=2$, then ...
7
votes
1
answer
358
views
Do i.i.d. sums concentrate any faster than martingales?
Suppose $X_1,X_2, \ldots, X_N \in \mathbb R^d$ are random variables with each $\|X_n\|_2 \le 1/2$ (this choice of the constant simplifies later formulae).
The simplest concentration inequality I know ...
13
votes
2
answers
1k
views
A comprehensive list of random walk inequalities?
I am interested in finding a comprehensive list of all noticeable random walk inequalities.
ie. $S_n = \sum_{k\leq n} X_i$ for i.i.d symmetric $X_i$
I can only seem to find books/papers that list ...