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.
9,022 questions
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Poisson process subordinated by a gamma process
I am working on a problem and I encountered the following situation:
$(N(t): t \ge 0)$ is a Poisson process with parameter $\lambda t $. If $T_{n} = \sum_{i=1}^n W_i$ represents the $n^\text{th}$ ...
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Arrangements of fixed $k$-polyplets in a $n\times n$ matrix
Recently, I asked a question about the number of arrangements of $k$ elements inside a $n\times n$ matrix with certain restrictions. The one I´m actually interested in for this question is in its 2. ...
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How large can $\mathbf{P}[X_1 + X_2 + X_3 < 2 X_4]$ get?
Let $\mu$ be a probability measure on $[0,\infty)$ and $X_1, \dots, X_4 \sim \mu$ independent. Then what can be said about the probability that $X_1 + X_2 + X_3 < 2 X_4$?
More precisely, what is ...
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Random covering on rectangles
Let $\mathrm{Rect}$ denote the class of axis-parallel rectangles $r: \mathbb{R}^2 \to \{0,1\}$, assigning $1$ if the point is inside the rectangle and $0$ otherwise. Let $\mathcal{D}$ be a ...
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Number of ways to place 4 kings on nxn chessboard
I have a $n\times n$ chessboard and 4 kings inside it. My goal is to count the number of arrangements where some of them are non-attacking or mutually attacking, for example:
In the case where the $4$...
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Coin toss stochastic question
Question:
There are $n$ people playing a game. Initially everybody had one dollar at hand. During each round of the game, we randomly pick two people and they will toss a fair coin, to decide who wins ...
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Count arrangements with pairs of attacking kings [closed]
I have a $1\times n$ chessboard and $2$ pairs of kings in it. Both components of each pair of kings must be adjacent in the chessboard, that is, they must be attacking.
Now, I want to calculate the ...
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Bound for the Malliavin derivative
Recently, I read the article Quantitative normal approximations for the stochastic fractional heat equation and I have a question in proof of Lemma 5.3. By using Lemma 5.3, they got
$$||D_{s,y}u(t,x)|...
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Mean of probability distribution
I have a probability distribution defined by the following density function:
$f(k,j,n,m)=\frac{(m n)! \mathcal{S}_k^{(j)}}{(m n)^k (m n-j)!}$ (With $\mathcal{S}_k^{(j)}$ being the Stirling number of ...
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Skorokhod-like construction for sequences of random probability measures
Let $(X_i)$ be a sequence of i.i.d. random vectors with distribution $P$ on $[0,1]^d$.
Let $D \equiv D([0,1]^d)$ be the multivariate Skorokhod space, equiped with a metric $d$ that makes it Polish. ...
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Average cluster size of a n-size vector
Given a vector of $n$ cells and $k$ elements in it, we can define a cluster of elements as a contiguous sequence of elements inside the vector.
My goal is to calculate the average cluster size for all ...
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Counting matrix paths for (n,m>2) matrices
Given a $n\times m$ matrix with $k$ elements inside it, I need to calculate the number of arrangements of those $k$ elements that form at least 1 path from the top to bottom matrix row composed of the ...
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$p$-norm of random variables and weighted $L^p$ space resemblance
I noticed a very similar relationship between weighted $L^p$ space (denoted $L_w^p$) and normed vector space of random variables. I want to unify these two spaces but there always seems to be a ...
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Kolmogorov complexity of Brownian motion
Given a 2 dimensional Brownian motion path $B\subset \mathbb{R}^2$, I am trying to show that for any oracle $A\subset\mathbb{N}$, there is a point $x \in B$ such that
$$\liminf_{r\to\infty}\frac{K_r^A(...
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Towards Schauder estimates: smoothing effect of the semi-group generated by $\Delta+(-\Delta)^{1/2}$
Consider the semi-group $(P_r)_r$ generated by $\Delta+(-\Delta)^{1/2}:$ for a distribution $f$ let $P_rf:=p(r,\cdot)*f$ where $p(r,x):=\sum_{q \in \mathbb{Z}^d}e^{2\pi\mathrm{i}\langle q,x\rangle}e^{-...
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Inference for the normal distribution with known variance from multiple clusters
Here's the question:
We have: $q \sim N\left(q_p, \frac{1}{\tau}\right), q_i \sim N\left(q, \frac{1}{\zeta}\right), t_n \sim N\left(0, \frac{1}{\eta}\right)$. Let $$ r_n=\sum_{i=1}^{\theta k_{n}} \...
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When is a family of distributions "closed" with respect to minimal sufficient statistics?
As in the title, I am interested in understanding how to express the idea that a parametric family of distribution is "closed" with respect to minimal sufficient statistics. Before giving ...
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Seeking strong bounds on KL-divergence and martingales for a hypothesis-testing inequality
Let's say we have a finite set $\mathcal{O}$ of observations, and let $\mathcal{C}(\Delta\mathcal{O})$ denote the space of closed convex sets of probability distributions.
We have two hypotheses which ...
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Conditional Expectation of Normal Distribution $E(q+t_1|r)$
I have difficulty deriving the follow conditional expectation:
there are N cluster of $q_{ni}+t_n$, each cluster has $k_n$ elements, $q_{ni}\sim N(q,\dfrac{1}{\zeta})$, $q\sim N(q_p,\dfrac{1}{\tau})$, ...
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Duplicating Matryoshka dolls
We start with a single doll of size $1$. Every second, independently of each other, every doll present produces a new doll of half its size with probability $\frac{1}{2}$. What is the expected size of ...
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Distributions of random walks on boundaries of balls in hyperbolic metric spaces
Suppose $G$ is a finitely-generated non-elementary hyperbolic group and consider a symmetric random walk on the Cayley graph $\text{Cay}(G,S)$ with generating set $S$. Denote the set of points $B_{\...
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Is the norm of first or second level of of signature a convex function?
I understand this is not a research level question but I really want to know, would anyone please help.
This question is related to the signatures that arises in rough path theory. https://en....
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Why do we need to define a random variable as a function?
I recently learned the mathematical definition of a random variable, namely:
A random variable is a measurable function $X: \Omega \rightarrow \mathbb{R}$ whose domain $\Omega$ is equipped with a $\...
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Hermite polynomial and Gaussian random variable
The following formula is well known: $E[H_k(X,E[X])H_q(Y,E[Y])]=\delta_{kq}E[XY]^k$ for a joint Gaussian r.v. $(X, Y),$ $H_k$ are Hermite polynomiale.
Is there a generalization for this to a joint ...
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Does stochastic boundedness imply stochastic domination by a constant multiple?
Let $X, Y$ be non negative random variables with finite expectation. We say that $Y$ stochastically bounds $X$ if there exists some $C > 0$ such that for all $x \in \mathbb R$,
$$\mathbb P(X \geq x)...
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Does stochastic domination of $X$ and $Y$ imply stochastic domination of $X \cdot Y$?
Suppose the random variables $X \geq 0$ and $Y \geq 0$ are both stochastically dominated by $Z \geq 0$, i.e.
\begin{align*}
& P(X \leq x), P(Y \leq x) \geq P(Z \leq x) \ , \ \forall x \geq 0 \ .
\...
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Towards the KPZ: Wiener-Ito integral, Kolmogorov type criterion
Consider a space-time white noise $\xi$ and the heat semi-group $(P_r).$
The following Kolmogorov type criterion allows to construct modifications in Besov Space (Here we have a partition of unity $(\...
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Rigorous analysis of phase transitions and universality in a non-linear model of interacting oscillators
Consider a system of interacting non-linear oscillators governed by the McKean-Vlasov equation:
$$\frac{\partial p(x,t)}{\partial t} = \frac{\partial}{\partial x}\left[\frac{\partial V(x)}{\partial x}...
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Weighted squared norm of multivariate truncated normal vector
Let $X \sim \mathcal{N}(0, \Sigma)$ be a multivariate normal vector with zero mean and inverse covariance matrix
$$
\Sigma^{-1} = \begin{pmatrix}
n & 1 & 1 & \cdots & 1 &...
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Wishart matrices: are eigenvalues and eigenvectors independent?
Let $W = X^TX$ denote a standard Wishart matrix, i.e., where $X$ is a Gaussian random matrix with iid standard Normal entries.
In this case we can write $W = U D U^T$, where $U$ is orthogonal and $D$ ...
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Interpretation of Lévy process with signed Lévy measures
Suppose that I have a non-decreasing, pure jump Lévy process of finite variation $X$ with Lévy measure $\pi$. The Lévy measure is then supported on $(0,+\infty)$. Suppose that the Lévy measure is a ...
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How to prove: $\gamma^2=\frac{n-p}{(n-1)p}\tau^2\sim F_{p,n-p}$, where $\tau^2\sim T^2(p,n-1)$
In multivariate statistics it is used to do hypothesis tests for Hotelling's $T^2$ distribution, but no textbooks prove this. Is there any proof for it?
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Questions about Lamperti's criteria for stochastic process recurrence
I'm working through Lamperti's 1960 paper "Criteria for the recurrence or transience of stochastic process. I" (J. Math. Anal. Appl. 1(3–4), 314–330. DOI: 10.1016/0022-247x(60)90005-6) as ...
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Limiting value of Stieltjes transform of sum of independent Wishart matrices
Let $n_1$, $n_2$, and $d$ positive integers tending to infinity such that $d/n_k \to \phi_k \in (0,\infty)$ and $n_1/(n_1+n_2) \to p \in (0,1)$. Let $X_k$ be an $n_k \times d$ random matrix with iid ...
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Approximating the probability that two Binomial variables are equal
Let $X,Y\sim Bin(n,p)$ be independent R.V.s and let $z\in[n]$ be integer.
My goal is to approximate the probability that $P[X-Y=2z]$.
What i need is a tight enough bound with error that is at most $o(\...
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Optimization over Poisson-binomial distributions
I am studying the problem of how an expected utility maximizer should optimally form a portfolio of uncorrelated Bernoullis.
Fix an increasing sequence of $n$ numbers in $(0,1)$, $0<p_1<\dots<...
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Law of large numbers for a continuum of Bernoullis
Suppose I have a family of $n$ independent Bernoulli random variables described by a vector of parameters $(p_i)_{i=1}^n$. As it is well known, the number of successes within this family is a random ...
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Exponential decay for Bernoulli percolation with a ghost field
Background on Bernoulli percolation:
Consider the hypercubic lattice $\mathbb{Z}^d$ as a graph with vertices as point of integer coordinates and edges between points of distance 1.
Now, delete every ...
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A question concerning regularly varying functions
In my work I need some results about regulary varying functions, which I only have a very vague understanding.
A strongly related reference I found is "On the Existence of a Regularly Varying ...
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Lower bound on the Lévy-Prokhorov metric for normal distributions
Let $\mathfrak M(\mathbb R^n)$ denote the metric space of probability measures (over $\mathbb R^n)$ equipped with the Lévy-Prokhorov metric $\rho$. Consider two $n$-variate normal distributions $\...
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Book recommendation in functional analysis and probability
I am interested by functional analysis and probability. I would like to know if you have any books that deal with these two subjects (at a graduate level) to recommend?
I'm looking for a book that has ...
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Looking for a counterexample: Conditioning increases regularity?
Let $p(x,y,z)$ be a joint density (over $\mathbb{R}^3$) under no smoothness or regularity assumptions, besides its existence. I am looking for a (counter)example where $p(y|x)$ is less regular than $p(...
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MGFs of sum of (Rademacher) independent variables and (hyperbolic/spherical) Pythagorean theorem
Consider a set of iid random variables $X_1, X_2, \ldots$ (distribution to-be-specified later). For real numbers $a_1, a_2, \ldots$ (with $\sum_{k} a_k^2 < \infty$) define
$$X = a_1 X_1 + a_2 X_2 +...
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Theories for "fuzzy" distributions
When calculating the probability density function for the quotients of adjacent values in an empirical time series, the image of the PDF looked like this:
It seems to resemble a lognormal ...
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A martingale puzzle about sum of expected squared bounds
I'm trying to get one of those "with $1-\delta$ probability, the following holds"-style bounds, and the following martingale problem looks solvable by some Freedman or Bernstein-style bound, ...
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Continuity dependence and convergence of the renormalized $\Phi^4_2$ model
This question is continuous for the one asked here: Local solutions of renormalized stochastic PDE but it was better to ask it separetely.
Again, we are interested in the local behavior of the $\Phi_2^...
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A rearrangement majorant of two random variables
$\newcommand{\Om}{\Omega}\newcommand{\F}{\mathcal F} $Let $X$ and $Y$ be random variables (r.v.'s) defined on a non-atomic probability space $(\Om,\F,P)$ such that $P(X<0)>0$ and $P(Y<0)>0$...
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Dependence and $L^2$ projections of functions
tl;dr: Is it possible that the best approximation to a nonnegative function of three variables with a bivariate function is no better than the best univariate function?
Let $w$ be a density on $\...
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131
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Almost sure convergence of double averages of IID random variables
Let $ \{X_i\}_{i=1}^{P} $ and $ \{Y_j\}_{j=1}^{Q} $ be two sequences of independent and identically distributed (i.i.d.) random variables. $X_i$ and $Y_j$ are independent between all pairs of $i$ and $...
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Expectation of maxima of random functions
I have a question regarded the expected maxima of random functions, but it seems easier to phrase using n-tuples of random variables:
Let $(X_{1}, X_{2}, ..., X_{n})$ and $(Y_{1}, ..., Y_{n})$ be ...
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