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If $X$ and $Y$ are random variables, then a maximal coupling of $X$ and $Y$ is a coupling $\left(X', Y'\right)$ such that $\mathbf{P}\left(X'=Y'\right)$ is maximal (that is, the probability that the c …

Given $\lbrace Y_i\rbrace$ a non-homogenous Poisson process with mean density $\theta y^{-1}e^{-y}$ where $y>0$ $(\theta>0)$. I.e., the number of points of $\lbrace Y_i\rbrace$ in $(a,b)$ with $0<a<b …

Given a fixed positive integer $n$, I have two random variables $$A(n)=2^{A_2}\cdots p^{A_{p_n}}, B(n)=2^{B_2}\cdots p^{B_{p_n}},$$ where $p_n$ is the largest prime number not exceeding $n$, $(A_p)_{p …

Let $X,Y$ be two discrete random variables. Two joint mass distributions (couplings) with marginals $X$ and $Y$ and with entries $p_{i,j}=\mathbb{P}_1(X=i,Y=j)$ and $p_{i,j}'=\mathbb{P}_2({X=i,Y=j})$ …

I have two discrete (integer-valued) random variables $A,B$, with $1\le A\le n$ and $1\le B$. A coupling is a joint distribution of $A,B$ with marginal distributions $A,B$. I know there are several wa …

I am learning about local limit theorems. The following example is probably why we don't have a "convergence in density/pmf." Ex: $X_1,X_2,\ldots$ is a sequence of independent RVs with mean $a$ and v …

In "Anatomy of Integers and Permutations", http://www.dms.umontreal.ca/~andrew/PDF/Anatomy.pdf, Granville gives a calibration of cycles of a permutation and prime factors of an integer. "We know rough …

It is known that Stein operator characterizes a probability distribution and there are a lot of ways of find a Stein operator. For example, if $Z$ is the standard normal distribution, with pdf(probab …

Given a positive integer $n$, let $p$ be the largest prime less than or equal to $n$. Let $N(n)=2^{C_2}\cdots p^{C_p}$ be uniformly distributed from $1$ to $n$, and $M(n)=2^{Z_2}\cdots p^{Z_p}$ where …