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Simple record times. It is well known that the record times have infinite expec- tation for a sequence of univariate iid rv with a common continuous df. This is no longer true in the multivariate case. In this section we give a precise characteriza- tion. 

Let X$_{1}$ , X$_{2}$ , . . . be i. i d. rv in R d with common continuous df F . We denote by N ( n ), n ≥ 1, the simple record times, i.e., those subsequent random indices at which a simple record occurs. Precisely, N (1) = 1, as X$_{1}$ is, clearly, a simple record, and, for n ≥ 2, 

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As the df F is continuous, the distribution of N ( n ) does not depend on F and, therefore, we assume in what follows without loss of generality that F is a copula C on R $^{d}$, i.e., each component of X$_{i}$ is on (0 , 1) uniformly distributed. 

Conditioning on X$_{1}$ = u yields for j ≥ 2 

Solving the geometric series, we get 

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Now we generalize this formula. Choose n ∈ N . Partitioning the sample space in disjoint events, we obtain for k$_{n}$ ≥ 1 

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