I just came across the following remark: If $(B_t)_{t\geq0}$ is a one dimensional Brownian motion and if we have a subdivison $0=t_0^n<…<t_{k_n}^n=t$ such that $\sup_{1\leq i\leq k_n}(t_i^n-t_{i-1}^n)$ converges to $0$ is $n$ converges to $\infty$ then $\lim_{n\to\infty}\sum_{i=1}^{k_n}(B_{t_i^n}-B_{t_{i-1}^n})^2=t$ in $L^2$. If $\sum_{n\geq 1}\sum_{i=1}^{k_n} (t_i^n-t_{i-1}^n)^2<\infty$, then we also have almost sure convergence Now I’m trying to find […]

Let $(X_n)_{n∈\mathbb{N}}$ a sequence of i.i.d. random variables uniformly distributed on the interval $[0, 1]$. Show that $$\limsup_{n\to+\infty} \frac{X_{2n}}{X_{2n+1}}=+\infty$$ a.s. I tried something that first I thought it was correct, but late I realized that maybe it is not. I tried to think about the random variable $\frac{X_{2n}}{X_{2n+1}}$ and use Borel-Cantelli to prove that the […]

In a proof of the Borel-Cantelli lemma in the stochastic process textbook, the author used the following. $$\limsup_{n\to\infty}A_n=\bigcap_{n\ge1}\bigcup_{k\ge n} A_k$$ Can someone explain why lim sup is intersection and union? Thank you

Let $(A_n)$ be a sequence of independent events with $\mathbb P(A_n)<1$ and $\mathbb P(\cup_{n=1}^\infty A_n)=1$. Show that $\mathbb P(\limsup A_n)=1$. It looks like the problem is practically asking to apply the Borel-Cantelli. Yet the suggested solution went differently: via $\prod_{n=1}^\infty \mathbb P( A_n^c)=0$. How can we apply the Borel-Cantelli lemma here? I.e. how to show […]

I was wondering what is the relation between the first and second Borel–Cantelli lemmas and Kolmogorov’s zero-one law? The former is about limsup of a sequence of events, while the latter is about tail event of a sequence of independent sub sigma algebras or of independent random variables. Both have results for limsup/tail event to […]

Let $\mathfrak{F} = (A_n)_{n \in \mathbb{N}}$. Prove that $P(\limsup A_n)=1$ if $\forall A \in \mathfrak{F}$ s.t. $P(A) > 0, \sum_{n=1}^{\infty} P(A \cap A_n) = \infty$. (Side question 1 Is second Borel-Cantelli out because we don’t know if the $A_n$’s are independent?) Suppose $\forall A \in \mathfrak{F}$ s.t. $P(A) > 0, \sum_{n=1}^{\infty} P(A \cap A_n) = […]

Let $\left\{ X_{n}\right\} _{n\in\mathbb{N}}$ be a sequence of independent random variables. Prove that $X_{n}\overset{a.e.}{\rightarrow}0$ if and only if, for all $\epsilon>0$, $\sum_{n=1}^{\infty}\textrm{P}\left(\left|X_{n}\right|>\epsilon\right)<\infty$. I guess I have to use the Borel Cantelli Lemma, right? Thanks.

Let $(\Omega,\mathcal F,P)$ be a finite measure space. Let $X_n:\Omega \rightarrow \mathbb R$ be a sequence of iid r.v’s I need to prove that if: $ n^{-1}\sum _{k=1}^{n} {X_k} $ converges almost surely to $Y$ then all $X_k$ have expectation. If I understand correctly then $X_k$ has expectations means $X_k$ is in $\mathcal L^1(\Omega)$. And […]

If $P(A_n) \rightarrow 0$ and $\sum_{n=1}^{\infty}{P(A_n^c\cap A_{n+1}})<\infty$ then $P(A_n \text{ i.o.})=0$. How to prove this? Thanks.

Let $W_0, W_1, W_2, \dots$ be random variables on a probability space $(\Omega, \mathscr{F}, \mathbb{P})$ where $$\sum_{k=0}^{\infty}P(|W_k|>k) <\infty$$ Prove that $$\limsup \frac{|W_k|}{k} \le 1 \ \text{a.s.} $$ I initially thought the conclusion meant $(**)$ when it really means $(*)$: $$P(\color{red}{(} \limsup |W_k|/k\color{red}{)} \le 1) = 1 \ \text{(*)}$$ $$P(\limsup \color{red}{(}|W_k|/k \le 1\color{red}{)}) = 1 \ […]

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