# Extreme value theory - proof this is a poisson point process

Let $(X_n)_{n \geq 1}$ be an i.i.d sequence of real valued RVs with continous distribution function f and $M_n:=\max \{X_1,...,X_n \}$. Let $U_n:=\inf \{ k \in \mathbb{N} | X_k>X_{U_{n-1}} \}$ be the n-th "record time" (with $U_0=1$)
Define (simple) point processes $N_n:= \sum_{k \geq 1} \delta_{U_k/n}$.(with delta being the dirac measure)
Are these $N_n$ poisson point processes? If so, what is their intensity measure.

So far I've shown that for disjoint sets $(A_k)$ $(N_n(A_k))$ are independent for all n, but im still missing the distribution property

• Out of curiosity, where did you find this problem? – fourierwho May 28 '18 at 16:55
• Its part of a Theorem in the german book "Einführung in die Extremwertstatistik" by Pfeiffer. The author gives a proof that $N_n$ converge locally weakly to a poissont process N with intensity measure 1/t dt on $(0,\infty)$. He proofs that the intensity measures of $EN_n(A) \rightarrow EN(A)$ for some Intervalls A. But for his proof to hold the N_n have to be Poisson Point Processes (at least according to the theorems he uses) – StefanWK May 28 '18 at 17:44