Let $\{X_i\}_{i=1}^\infty$ be a sequence of i.i.d. random variables with a continuous CDF.
Let $V_1:=\min \{n\in\mathbb N\, \vert \,X_n>X_1\}$.
Let $V_{r+1}:=\min \{ n\in \mathbb N \,|\, n>V_r \mbox{ and } X_n>X_{V_r} \}$ for all $r\ge 1$.
Define $\Delta_1 :=V_1$ and $\Delta_{r+1}:=V_{r+1} -V_r$ for all $r\geq 1$.
Now I want to show that $$\frac{\log \Delta_r}{r}\to 1 \,\,\mbox{ almost surely.}$$
This is supposed to be easy because it is in the warmups of a sequence of exercises, but I could not think of a easy way to solve it.
I can prove the convergence above by proving that $\Delta_r \to e^r$ almost surely. However, the proof is quite long and will use many external results. Actually the first proof of that $\Delta_r \to e^r$ almost surely was published in the 60's in a journal.
The desired inequality above shall be much weaker and easier to be proved but I just couldn't think of an easy way... Thanks for any comment/hint/answer.
As the CDF is continuous, we may assume the $X_i$ are $U[0, 1]$. Set $P_0 = 1$, $P_r = 1 - X_{V_r}$, and $Q_r = P_r/P_{r-1}$. The $Q_{r}$ are i.i.d. $U[0, 1]$, so using the SLLN, $$\frac{\log P_r}{r} = \frac1r \sum_{i=1}^r \log Q_i \overset{a.s.}\to -1;$$ as a consequence, we also obtain $P_r \overset{a.s.}\to 0$.
We further have that $\Delta_{r+1} \mid P_{r} \sim \mathrm{Geo}(P_{r})$, so using Borel-Cantelli, $$\frac{\log \Delta_{r+1} + \log P_{r}}{r}\overset{a.s.}\to 0.$$ The result follows since $\frac{r}{r+1} \to 1$.