26 CL ´ EMENT DOMBRY, MICHAEL FALK, MAXIMILIAN ZOTT and further, similar to the calculation above, Hence, solving all the occuring geometric series yields Furthermore, it is easy to see that for all k ≥ 1 which means that N ( n + 1) − N ( n ) | X$_{N}$$_{(}$$_{n}$$_{)}$ = u is geometrically distributed with parameter 1 − C ( u ). We summarize these results. Lemma 4.5. Let X$_{1}$ , X$_{2}$ ,... be iid rv following a copula C on [0 , 1] $^{d}$, and denote by N ( n ) the n -th simple record time, n ∈ N . where Geom( p ) denotes the geometric distribution with support { 1 , 2 ,... } and parameter p ∈ (0 , 1] . Suppose now that d = 1. Then we have u = u ∈ [0 , 1], C ( u ) = u and which is well-known (Galambos (1987, Theorem 6.2.1)). N ( n + 1) − N ( n ) | X$_{N}$$_{(}$$_{n}$$_{)}$ = u ∼ Geom(1 − C ( u )) , (17) P ( N ( n + 1) − N ( n ) = k | X$_{N}$$_{(}$$_{n}$$_{)}$ = u ) = C ( u ) k − $^{1}$(1 − C ( u )) , E ( N ( n + 1) − N ( n )) = ∫ { u $_{n}$̸≤···̸≤ u $_{1}$} n ∏ i =1 1 1 − C ( u$_{i}$ ) C (d u$_{1}$ ) · · · C (d u$_{n}$ ) . P   N ( n + 1) = n ∑ j =1 k$_{j}$,N ( n ) = n − 1 ∑ j =1 k$_{j}$,...,N (2) = k$_{1}$   = ∫ { u $_{n}$̸≤···̸≤ u $_{1}$} C ( u$_{1}$ ) k$_{1}$ − $^{2}$C ( u$_{2}$ ) k$_{2}$ − $^{1}$· · · C ( u$_{n}$ ) k$_{n}$ − $^{1}$(1 − C ( u$_{n}$ )) C (d u$_{1}$ ) · · · C (d u$_{n}$ ) . (i) For every n ∈ N (ii) For every n ∈ N , E ( N (2)) = ∫ 1 0 1 1 − u du + 1 = ∞ , (18) E ( N ( n + 1) − N ( n )) = ∫ { u $_{n}$̸≤···̸≤ u $_{1}$} n ∏ i =1 1 1 − C ( u$_{i}$ ) C (d u$_{1}$ ) · · · C (d u$_{n}$ ) .