Let us consider a random walk $(S_n)_n$. One denotes the instants of records of $-S_n$ by $0=T_0 < T_1 < T_2 \cdots$. Then for all $k$ one sets: $H_k=-S_{T_k}$. Finally, one define $\tau$ as the first positive time at which the random walk is above $0$.
Then, one has, for all nonnegative $x$: $$ \sum_k P(H_k \le x) = E \sum_{j \le \tau-1} 1_{-x \le S_j}. $$
This is seemingly a very basic fact but the only indication in the paper I read is "duality lemma" (and I do not know what it refers to). Does anyone has a precise reference or could anyone provide a sketch of a proof ?
Thank you.
