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Let $X$ be a random variable with $E[X] = \mu < \infty$. For $n=1,2,\dots$, construct a triangular array of random variables as \begin{equation} Y_{n,i} = X_i \frac{\sqrt{\mu}}{\sqrt{\sum_{j=1}^n X_ …

I have a random variable $X$ defined on $(0,\infty)$. For each $n\in \mathbb N$, define $X_n = X \mathbf{1}_{0 < X \leq C_n}$, where $C_n$ is a monotonically increasing sequence of positive numbers su …

Let $\{X_n\}_{n\in \mathbb{N}}$ be a sequence of nonnegative discrete random variables, and $X$ be a nonnegative discrete random variable. The probability generating function $\psi_X(z)=\sum_{k=0}^\in …

Let $\{X_n\}_{n=1}^\infty$ be a sequence of random variables such that, $$ \forall n,\,\,\mu_n := EX_n < \infty, \quad \mu_n \to \mu < \infty \textrm{ as } n\to\infty. $$ Now for each $n$, let $X_{n …

I have a random variable $X$ whose first and second moments are given as $$ E[X] \propto C_n^{1-a},\quad E[X^2] \propto C_n^{2-a}\quad (0 < a < 1), $$ where $C_n$ satisfies $$ \lim_{n\to\infty} C_n = …