30 CL ´ EMENT DOMBRY, MICHAEL FALK, MAXIMILIAN ZOTT But this implies that the above integral is finite and, thus, the assertion is a conse- quence of Lemma 4.6. □ Next we investigate E ( N ( n )) for n ≥ 2. As before, let X$_{1}$ , X$_{2}$ , · · · be an iid sequence of rv on R d following a copula C . Obviously, E ( N (2)) = ∞ implies E ( N ( n )) = ∞ for n ≥ 2, since N ( n ) ≥ N (2), n ≥ 2. On the other hand, if E ( N (2)) < ∞ , we obtain due to (18) for all n ≥ 2 which means that E ( N ( n + 1) − N ( n )) < ∞ as well in that case. Furthermore, we will show below that E ( N ( n + 1) − N ( n )) = ∞ for all n ∈ N if E ( N (2)) = ∞ . In conclusion, it is sufficient to decide whether E ( N (2)) is finite or not if expecations of arbitrary simple record times are investigated. We summarize this discussion. Proposition 4.10. Let X$_{1}$ , X$_{2}$ ,... be iid rv following a copula C on [0 , 1] $^{d}$. Then the following implications hold: Proof. It remains to proof (ii). We show that N ( n + 1) − N ( n ), n ∈ N , is stochastically increasing, i. e. for every t ∈ R and every n ∈ N Recall that the df of a rv X ∼ Geom( p ) is given by P ( X ≤ t ) = 1 − (1 − p ) ⌊ t $^{⌋}$, where ⌊ t ⌋ = max { m ∈ Z : m ≤ x } . Conditioning on X$_{N}$$_{(}$$_{n}$$_{)}$ = x , we obtain by (21) P ( N ( n + 1) − N ( n ) ≤ t ) ≥ P ( N ( n + 2) − N ( n + 1) ≤ t ) , t ∈ R , n ∈ N . E ( N ( n + 1) − N ( n )) = E ( 1 1 − C ( X$_{1}$ ) · · · 1 1 − C ( X$_{n}$ ) 1$_{{}$ X $_{n}$̸≤···̸≤ X $_{1}$} ) ≤ E 1 1 − C ( X$_{1}$ ) · · · 1 1 − C ( X$_{n}$ ) ) ≤ [ E ( 1 1 − C ( X$_{1}$ ) )] n = [ E ( N (2)) − 1] $^{n}$, (i) If E ( N (2)) = ∞ , then E ( N ( n + 1) − N ( n )) = ∞ for all n ∈ N . (ii) If E ( N (2)) < ∞ , then E ( N ( n + 1) − N ( n )) < ∞ for all n ∈ N .