I have a series of continuous random variables $X_t$, $t \geq 0$, that satisfy the following two asymptotic relations:
- For any $n \geq 0$ and any $x$, let $t_n(x) \geq 0$ be the time corresponding to the $n$th local maximum of $X_t(x)$. Then there exists a sequence of random variables $Y_n$ such that $\lim_{n \to \infty} \frac{X_{t_n}(x)}{Y_n(x)} = 1$ for a.e. $x$.
- For a.e $x$, $\lim_{n \to \infty} \frac{t_n(x)}{cn} = 1$.
Do we necessarily get that $$\lim_{T \to \infty} P(\max_{1 \leq t \leq T} X_t(x) \leq T) = \lim_{N \to \infty} P(\max_{1 \leq n \leq N} Y_n(x) \leq cN)?$$ Do we need extra assumptions for an identity like this to be true?