content_image/325128.jpg content_image/325127.jpg content_image/325126.jpg content_image/325132.jpg content_image/325125.jpg content_image/325131.jpg
and further, similar to the calculation above, 

<img src='content_image/325128.jpg'>

Hence, solving all the occuring geometric series yields 

<img src='content_image/325127.jpg'>

Furthermore, it is easy to see that for all k ≥ 1 

<img src='content_image/325126.jpg'>

which means that N ( n + 1) − N ( n ) | X$_{N}$$_{(}$$_{n}$$_{)}$ = u is geometrically distributed with parameter 1 − C ( u ). We summarize these results. 

Lemma 4.5. Let X$_{1}$ , X$_{2}$ ,... be iid rv following a copula C on [0 , 1] $^{d}$, and denote by N ( n ) the n -th simple record time, n ∈ N . 

(i) For every n ∈ N 

where Geom( p ) denotes the geometric distribution with support { 1 , 2 ,... } and parameter p ∈ (0 , 1] . 

(ii) For every n ∈ N , 

<img src='content_image/325132.jpg'>

Suppose now that d = 1. Then we have u = u ∈ [0 , 1], C ( u ) = u and 

which is well-known (Galambos (1987, Theorem 6.2.1)). 

<img src='content_image/325125.jpg'>

<img src='content_image/325131.jpg'>

