Let $g^k$ be a sequence of processes satisfying $$P\left(\int_0^T |g^k-g^{k+1}|^2\,ds>2^{-k}\right) \leq 2^{-k}$$

Then the book I am reading states that as a application of the Borel-Cantelli lemma we get that

$$\sum_{k=1} ^ \infty \left(\int_0^T |g^k-g^{k+1}|^2\,ds\right)^{1/2}< \infty \quad \text{a.s.} \quad P.$$

Can anyone please explain how Borel-Cantelli Lemma is being used to arrive at this .


By the Borel-Cantelli Lemma, you can get that $$ \mathbb{P}(\limsup_{k\to\infty} \{ \int_0^T |g^k - g^{k+1}|^2 ds\}>\frac{1}{2^k}\})=0 $$ This means that $\mathbb{P}$-a.s. there exists $N\in \mathbb{N}$ such that for all $k\geq N$: $\int_0^T |g^k - g^{k+1}|^2 ds\leq \frac{1}{2^k}$, therefore $(\int_0^T |g^k - g^{k+1}|^2 ds)^{1/2}\leq (\frac{1}{\sqrt{2}})^k$ it implies that $\sum_{k\in\mathbb{N}}(\int_0^T |g^k - g^{k+1}|^2 ds)^{1/2}<\infty$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.