ON FUNCTIONAL RECORDS AND CHAMPIONS 25 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, 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 Now we generalize this formula. Choose n ∈ N . Partitioning the sample space in disjoint events, we obtain for k$_{n}$ ≥ 1 P ( N ( n + 1) − N ( n ) = k$_{n}$ ) = = ∞ ∑ k$_{1}$ =2 ∞ ∑ k$_{2}$ =1 · · · ∞ ∑ k$_{n}$$_{−}$$_{1}$ =1 P   N ( n + 1) = n ∑ j =1 k$_{j}$, N ( n ) = n − 1 ∑ j =1 k$_{j}$, . . ., N (2) = k$_{1}$   , (16) E ( N (2)) = ∞ ∑ j =2 jP ( N (2) = j ) = ∫ [0 , 1] d 1 1 − C ( u ) C (d u ) + 1 . P ( N (2) = j ) = P ( X$_{2}$ ≤ X$_{1}$ , . . . , X$_{j}$$_{−}$$_{1}$ ≤ X$_{1}$ , X$_{j}$ ̸≤ X$_{1}$ ) = ∫ [0 , 1] d C ( u ) j − $^{2}$(1 − C ( u )) C (d u ) . N ( n ) := min { j : j > N ( n − 1) , X$_{j}$ ̸≤ max 1 ≤ i ≤ N ( n − $_{1)}$X$_{i}$ } .