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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 

<img src='content_image/306088.jpg'>

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: 

(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 . 

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 

<img src='content_image/306087.jpg'>

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 

