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This answer contains the details to the answer provided by Michael, all credits go to him. Choose $M>0$ such that $P(|X_t|<M)>1/2$ and $P(|Y_t|<M)>1/2$. Define $A_t = 1$ if $|X_t|<M$ and 0 otherwise. …

Suppose I have a two sided stationary sequence of random variables $\ldots,X_{-1},X_0,X_1,\ldots$ such that all finite dimensional joint densities $f(x_1,\ldots,x_n)$, $n\in\mathbb{N}$ exist. I want t …

A while ago I posted the following problem: Suppose I have a two sided stationary sequence of random variables $\ldots,X_{-1},X_0,X_1,\ldots$ such that all finite dimensional joint densities $f(x_1,\ …

Let $X_1,X_2,\ldots$ be a sequence of identically distributed random variables and let $A\subset\mathbb{R}$ be some set such that $P(X_1\in A)<1$. I have a product of indicator functions $$ \lim_{N\ri …

Let $\{X_t\}_{t\ge1}$ and $\{Y_t\}_{t\ge1}$ be two iid sequences of random variables that have full support. That is, if $A\subseteq\mathbb{R}$ has positive Lebesgue measure, then $P(X\in A) >0$ and $ …

Let $\{\epsilon_t\}_{t\ge0}$ be a sequence of iid random variables with full support. Let $\delta\ge0$ and $\gamma>1$. Then set $X_0 = \delta$ and define for $t\ge0$: $$ X_{t+1} = \gamma X_{t} + \epsi …

Setup: Suppose $X = \{X_n\}_{n\in\mathbb{Z}}$ is a stationary ergodic proces on the real line and let $A = \prod_{n\in\mathbb{Z}}A_n$ be a Borel measurable set such that $$ P(X \in A) = P\left(X_n\in …

Let $\{X_k\}_{k\in\mathbb{N}}$ be a stationary ergodic sequence on a probability space $(\Omega,\mathcal{F},P)$ with shift $T$. Also, let $\{v_k\}_{k\in\mathbb{N}}$ be a sequence of random variables i …

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of strictly positive and identically distributed random variables and let $\beta\le 1$. I am trying to prove that $$ 0<\lim_{\beta\rightarrow 1}(1-\beta …