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.
9,027 questions
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Exponential tail bounds without the moment generating function
As an exercise, I thought I would try to prove some classical Chernoff bounds without ever using the moment generating function, but then found myself getting stuck in certain places. Before I state ...
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Parametric vs Non-parametric Estimation of Quantiles
Motivation
Suppose that we need to estimate the median from a normal distribution with known variance. One non-parametric approach is to use the sample median as an estimator. However, this does not ...
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Commuting supremum and expectation
Given a one-parametric random function on a probability space $(\Omega,\mathcal F,\mathbb P)$:
$X:U\times\Omega\to \mathbb R \text{ and } (a,w)\mapsto X(a,w), \text{ with } \sigma(X(a))\subseteq \...
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Brownian Approximation of Downswings of Walks with Positive Drift
I'm interested in the downswings of discrete walks w(t) whose steps are IID, bounded, and have positive mean. A simple example might have steps which are +1 with probability 2/3, and -1 with ...
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Singular Fisher information matrix and existence of unbiased estimators
I'm doing some research into the Cramer-Rao bound for time of arrival localization and have come across a rather strange result: the FIM is singular, but there exists an unbiased estimator. My ...
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Why does non-decreasing entropy imply actual convergence to that max entropy distribution?
Let $X_n$ be i.i.d with finite variance. Let $\bar X_n=\frac 1n \sum_{i=1}^nX_i$. It is a famous result that the continuous/differential entropy of the normalized average is non-decreasing. $$\mathrm ...
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Projections in infinite dimensional statistical manifolds
I'm struggling to understand the geometry of projection for infinite dimensional statistical manifolds. In finite dimensions, a strictly convex smooth function $F$ defines a Bregman divergence. From ...
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Examples of probability measures with `fake' decay
To be concise, I am wondering whether there are natural examples of probability measures $\mu$ compactly supported on the real line which satisfy $\mu(I) \lesssim l_n^\alpha$ for all intervals $I$ ...
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Combinatorial/probabilistic statements having $F_{\text{un}}$/$F_q$ geometric interpetation
$\newcommand{\Fun}{F_\text{un}}$There was lots of "Fun with $\Fun$" (field with one element) in recent years.
One of the points is that it provides bridge between geometrical and ...
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Expectation involving maximum of Gaussian variables
Let $X\sim N(0, I_d)$ be a $d$-dimensional Gaussian random vector. Let $W_1, \ldots, W_k \in \mathbb{R}^d$ be $k$ fixed vectors in general positions. It is clear that $w_i^\top X, \ldots, w_k^\top X$ ...
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Winding number of a random walk on the square lattice before hitting the origin
Let us consider a simple random walk on $\mathbb{Z}^2$ started at $(x,0)$ and killed upon hitting the origin. Define the total winding number $w_x$ around the origin to be the (signed) number of ...
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approximate stationary distributions of a doubly stochastic matrix and its supports
Given a doubly stochastic matrix $M$ and a distribution $v$,let $M=\sum_{\sigma\in S_n}p_{\sigma}M_{\sigma}$ be any Birkhoff decomposition of $M$, where $M_{\sigma}$ is the permutation matrix induced ...
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Statistical independence and the Fourier transform
Consider the random variable $S=(s_0, \dots ,s_{N-1})$, a sequence of signs uniformly distributed on the hypercube $\{-1,1\}^N$. With the Fourier transform we can define $N$ random walk variables
$$
\...
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Another name for coin-flipping polynomials
In his paper Functions arising by coin flipping (section 4), Johan Wästlund coined the term "coin-flipping polynomial" for polynomials that arise in connection with observing a finite number of coin ...
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Convex conjugate of relative entropy
The convex conjugate of a function, say, $f:X\to \mathbb{R}$ is a function $f^*:X^*\to \mathbb{R}$ defined as
$$f^*(x^*):=\sup_{x\in X} ~\langle x, x^*\rangle-f(x),$$ where $X^*$ is the topological ...
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Closure of random rotations
Are matrix Fisher random variables closed under multiplication?
For those unfamiliar with the jargon, let me unpack the terms above and repose my question.
This is a question about probability ...
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Random Functions and Transition Probabilities
Let $(S,\mathcal{S})$ and $(T,\mathcal{T})$ be measurable spaces. A transition probability from $S$ to $T$ is a function $\pi:S\times\mathcal{T}\to [0,1]$ such that $\pi(s,\cdot)$ is a probability ...
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A simple decomposition for fractional Brownian motion with parameter $H<1/2$
Background
Let $X = \{X(t):t \geq 0\}$ be a (standard, real-valued) fractional Brownian motion (fBm) with parameter $H \in (0,1)$, i.e., a continuous centered Gaussian process with covariance ...
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The question about Kolmogorov tightness criterion
We know about Kolmogorov Criterion for the tightness of a stochastic process $X_n(t)$
1.The sequence $(X_{n}(0))_{n\geq0}$ is tight.
2.There exist constants $\gamma\geq0$,$\alpha>1$, $K>0$ and ...
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Random, Linear, Homogeneous Difference Equations and Time Integration Methods for ODEs
Most methods (that I know of) of numerically approximating the solution of ODEs are "general linear methods". For this type of method, the so-called 'linear stability' is examined by applying the ...
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Feynman-Kac formula for the GFF
The Feynman-Kac formula says that $$ \exp(-t(-\Delta+V(X)))(x,y) = \mathbb{E}_{\gamma(0)=x,\gamma(t)=y}\left[\exp(-\int_0^t V\circ\gamma)\right] $$ where $\Delta$ is the Laplacian on $L^2(\mathbb{R}^n)...
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Normal distribution by successive approximation?
$\newcommand\R{\mathbb R}\newcommand\la\lambda$It is well known and easy to see that the rotationally invariant
product of two probability measures on $\R$ has to be a Gaussian (or Dirac) measure; see ...
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Minimizing the largest eigenvalue of random matrices
Let $A \in \mathbb{R}^{n \times n}$ be a symmetric matrix with entries $A_{ij} \sim \mathcal{N} (0,1)$, all independent except for the symmetry condition.
Consider the following minimization problem:...
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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 $U^\top\sqrt{D^{-1}}(P+(\mathrm{e}^x-1)B(0)-\mu I)\sqrt{D}U$, ...
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Probabilistic characterization of first Neumann eigenvalue
In this MO post, a question has been asked (and answered) about the probabilistic interpretation of the first Dirichlet eigenvalue of the Laplacian in terms of boundary hitting times.
I wish to ask ...
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What is the six positive real number for a dice producing a highest chance?
Say there is a dice with six faces, each face has a positive real number different from others. There is a chessman on the origin of the number axis. In each trial, the dice rolls infinite times. ...
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Random pro-p groups via iterated uniformly random central extensions
Inspired by this question on math.se, I want to understand the following construction of a random pro-$p$ group:
We want to construct an inverse system
$$\cdots \xrightarrow{\alpha_i} G_i \...
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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 2....
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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 ...
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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 ...
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First Collision Time for k Random Walkers on a Torus
I consider $k$ random walkers on $\mathbb{Z}^{d}/n \mathbb{Z}^{d}$, the $d$-dimensional torus of side length $n$. More precisely, I will define a Markov chain $Z_{t} = (X_{t}[1], \ldots, X_{t}[k])$ ...
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Doob's inequality for martingale "convolution"
Let $(X_t, t \in \mathbb{N})$ be a martingale, and let $a \leq b \leq T \in \mathbb{N}$ be constants. Is there something like Doob's inequality for $\mathbb{E} \sup_{a \leq t \leq b} X_t(X_T-X_t)$, i....
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Random walk over a function
Let $\{X_n\}_{n\geq 0}$ be a random walk. Let us assume that $\mathbb{E}X_1 =0$ and $\mathbb{E}X_1^2=1$. Let also $\mathbb{E}\exp(c|X_1|)<+\infty$ for some $c>0$ and $X_1$ has a law with ...
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Tails of sums of Weibull random variables
Suppose that $X_1, X_2, \ldots, X_n$ are i.i.d random variables distributed according to Weibull distribution with shape $0 < \epsilon < 1$ (it means that $\mathbf{Pr}[X_i \geq t] = e^{-\Theta(t^...
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Is there a problem with the Wishart distribution?
It seems that the Wishart distribution scales problematically with dimension. I start with some background, which should make the question reasonably understandable to a general math and statistics ...
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Infimum of the Dirichlet form for a tensor product
If $Q$ is the generator of a well-behaved continuous-time Markov process on a finite state space and $p$ is the invariant distribution, the corresponding Dirichlet form is $\mathcal{D}_Q(f) := \frac{1}...
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Examples of Slowly Mixing Chains in Statistics
This should probably be community wiki, but I don't know how to set that myself.
I'm looking for examples or Markov chains that are used in statistics or statistical physics, and which are known to ...
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$\lim_{n \to \infty} E[V| W +\frac{1}{n}V ]$ where $W$ and $V$ are independent
Let $V$ and $W$ be independent random variables. Assume that $V$ is standard normal.
We are interested in the following limit
\begin{align}
\lim_{n \to \infty} E[V| W +\frac{1}{n}V ]
\end{align}
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Infinite cardinals and learnability of probability distributions
Two players play as follows. Player one chooses a secret finitely supported probability distribution $P$ on $ω_k$ (or another known set with $\aleph_k$ elements), and randomly takes $n+1$ samples ...
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Looking for the eigenfunctions of the operator $T$ on $L_2(\mathbb R^+)$ defined by $Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy$
I'm looking to find a basis of eigenfunctions (and the corresponding eigenvectors) for the operator $T$ on $L_2(\mathbb R^+)$ defined by:
$$
Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy
$$
This operator ...
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Why does this oddly nice lognormal shape come out of tree?
I was reading this paper and came across the Euclid tree. For a simple version of this tree, consider an infinite binary tree rooted with the triple $(1,2,3)$. Then for each vertex $(x,y,z)$, its left ...
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Growth of spheres in FINITE nilpotent groups - Gaussian approximation (central limit theorem)?
Standard setup. Consider a group and choose generators. Word-metric (or in the other words - distance on the Cayley graph of the group+generators) - converts a group into a metric space, which is ...
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The uniform odd and even subgraph of $\mathbb{Z}^2$
Given a (first finite and later infinite) graph $G =(V,E)$ the uniform even graph is the uniform probability measure on the set of spanning even subgraphs. That is subgraphs (V, E') with $E' \subset E$...
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Projected polar chessboard measure convergence in total variation?
$\newcommand\R{\mathbb R}\newcommand\C{\mathbb C}\newcommand\ga{\gamma}$For natural $n$, let $E_n$ be the set of all points in $\R^2$ with "polar coordinates" $(r,t)$ in the set
$$F_n:=\...
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Relationship between measure theory and quantification
I was advised that this question might be better suited for mathoverflow, so I am reposting it here (original post).
In a 1978 paper published by David P. Ellerman and Gian-Carlo Rota, the duo discuss ...
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Counting returns in null-recurrent random walk
Consider two independent copies of IID random walk on ${\bf Z}$ starting from $0$, and let $N_1(t)$ (resp. $N_2(t)$) denote the number of times, up to time $t$, that the first (resp. second) walker ...
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Mistakes in Logan and Shepp's famous paper on Young Tableaux?
In their landmark paper from 1977 named "A Variational Problem for Random Young Tableaux" Logan and Shepp obtained a number of results concerning asymptotic properties of Young Diagrams ...
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Approximating any convex shape in $\mathbb{R}^d$ with a polytope having $\mathrm{poly}(d)$ facets
We denote by $V(A)$ the $d$-volume of any convex set $A$. Furthermore, given any two convex sets $A,B\in\mathbb{R}^d$, we denote by $V_{A,B}$ the $d$-volume of the symmetric difference $V\left(A \...
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Sum of variables uniformly distributed on a circle: a cyclic property
Consider a random walk starting at the origin in the plane, walking $n$ steps in independent uniformly random directions with step lengths $a_1,\ldots,a_n$, and observing the distance from the origin. ...
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Is there a practically useful or concrete representation theory/Fourier analysis on finite groupoids?
Fourier analysis on finite groups is well known to be useful for probability theory and combinatorics — consider for example the Fourier analysis on $(\mathbb Z/2\mathbb Z)^n$ which can be used to get ...
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