Does very fast convergence in probability imply almost sur convergence for a continuous stochastic process?

Question

I was wondering if someone knows how to prove the following fact (which might not be a fact ;) ):

let X being a stochastic process with almost surely continuous sample path, and such that, there exists a constant $\eta>0$ such that,

$ \forall \epsilon>0,\ \exists \mathcal{C}_{\epsilon}>0, \ \mathbb{P}(|X_t|>\epsilon)\leq \mathcal{C}_{\epsilon} e^{-\eta t}. $

Then, $X_t$ converge almost surely to zero!

In discrete time, such question is easly solved using Borel-Cantelli Lemma. How to make it works in continuous time? Else, have you a counter-exemple? In all my trials, I wasn't be be able to prove this statement without using some kind of uniform continuity...

Thank you!

Answer 1

Consider a sequence of independent uniform(0,1) random variables $U_j$, and let $$X_t = \sum_{j=1}^\infty \max(0,1 - 2^j |t - j - U_j|)$$ In particular $X_t = 0$ outside the intervals $(j + U_j - 2^{-j}, j + U_j + 2^{-j})$, and so $P(X_t > 0) < 2^{2-t}$. But of course $X_t$ does not converge to $0$.