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$\newcommand\R{\mathbb R}\newcommand\la\lambda$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 …

Let $X,X_0,X_1,\dots$ be nonnegative independent identically distributed (i.i.d.) random variables. Let \begin{equation*} E:=\bigcap_{n\ge0}B_n, \end{equation*} where \begin{equation*} B_n:=\b …

$\newcommand\R{\mathbb R}$Let $(f_n)$ be a sequence in $L^1(\R)$ converging weakly to some $f\in L^1(\R)$. Let $(a_n)$ be sequence in $\R$ converging to $0$. For each natural $n$, let $g_n$ be the $a_ …

$\newcommand{\R}{\mathbb R}$Consider the following construction. For real $u$, let \begin{equation} f(u):=\frac{2u^2}{1+u^4}, \end{equation} so that the function $f\colon\R\to\R$ is continuous, $0 …

Let $(f_n)$ be a sequence of twice differentiable functions on $\mathbb R$ such that for each $n$ there exists some $x_n\in\mathbb{R}$ such that: $f_n$ is strictly convex on $(-\infty,x_n)$, $f_n$ is …

$\newcommand\R{\mathbb R}$For a probability measure $\mu$ over $\R^2$ and a unit vector $u\in\R^2$, let $\mu^u$ denote the pushforward of $\mu$ under the projection map $\R^2\ni x\mapsto u\cdot x\in\R …

$\newcommand{\R}{\mathbb R}$Let $M$ denote the set of all finite signed measures on a separable Banach space $B$. For any $\mu\in M$, let \begin{equation*} \exp^*(\mu):=\sum_{k=0}^\infty\frac{\mu^ …

The following question was asked in a comment at Almost sure convergence vs convergence of probability density functions : Suppose that $(X_n)$ is a sequence of random variables (r.v.'s) converging a …

$\newcommand\R{\mathbb R}$Let us say that a function $r\colon\R_+\to\R_+$ is a rate function if $r$ is nondecreasing and $r(x)\to\infty$ as $x\to\infty$. Let us say that a family $(r_j)_{j\in J}$ of r …