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Let $X_n, X$ be $[0, 1]$-valued random variables whose laws are absolutely continuous with respect to Lebesgue measure. Suppose $X_n \to X$ a.s. Does this imply that the pdfs of $X_n$ converge to that …

Write $\mathcal S$ for the set of probability measures on $[0, 1]$ that are non atomic and singular with respect to Lebesgue measure. Two measures $\mu$ and $\nu$ in $\mathcal S$ are said to be topolo …

Motivation: This is a toy model of how a closed system will always evolve towards the distribution of maximal entropy, where no further transfer of heat/energy is possible. Problem set up: Fix a posit …

Definitions: Two real valued random variables $X_0$ and $X_1$ are called a one step martingale if $E[X_1| X_0] = X_0$. We say the one step martingale is in $L^2$ if both $X_0$ and $X_1$ are in $L^2(P) …

Suppose $X, Y$ are $L^1$ random variables, and $X_t$ and $Y_t$ are real valued stochastic processes with $X_t, Y_t \in L^1$ for all $t$ such that the following convergences hold: i) $X_t \to X$, $Y_t …

Does there exist a non constant almost surely continuous stochastic process $X$ on $[0, \infty)$ with $X_t$ independent of $X_{t+1}$ for all $t \geq 0$?

Given a random variable $X$, we denote by $X_1, X_2, \dots$ a sequence of iid copies of $X$. Question: Does there exist a random variable $X$ with $\mathbb E[X^+] = \mathbb E[X^-] = +\infty$, but $$\l …

Let $X_n$ be a sequence of random variables with uniformly bounded $L^1$ norm. Suppose $X_n$ converges in probability to $X \in L^1$. Is it true that the Cesaro sums $Y_n := \frac{1}{n} \sum_{i = 1}^n …

There are $N \geq 1$ white balls and $1$ black ball in an (infinitely big) urn. Every turn, a ball is drawn from the urn uniformly at random. If a white ball is drawn, it is put back into the urn alon …

Two teams are having an intense penalty shootout. The game ends when either team leads by a certain threshold, or once a certain number of rounds has passed, whichever comes first. Currently team $X$ …

Fix some $0 < r < 1$. A collection of points $x_1, \dots, x_n$ are chosen independently and uniformly at random from the closed unit ball in $\mathbb R^d$. What is the probability that the intersectio …

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 …

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) \ …

Let $C > c > 0$ and $K > 1$ be constants. Does there exist, for all small enough $\varepsilon > 0$ depending on $c, C, K$, some bound of the following form? For all random variables $X$ with $c \leq …

Notation: We write $\mathbb E_{\mathcal F} X$ for the conditional expectation $\mathbb E[X|\mathcal F]$ of a random variable $X$ with respect to a $\sigma$-algebra $\mathcal F$. Let $X$ be an integrab …