Lemma 4.5 (i) for each t ∈ R and n ∈ N ON FUNCTIONAL RECORDS AND CHAMPIONS 31 which shows (21) since X$_{N}$$_{(}$$_{n}$$_{)}$ ≤ X$_{N}$$_{(}$$_{n}$$_{+1)}$ . Hence, Computing the distribution of the second record X$_{N}$$_{(2)}$ is an easy task: Let X$_{1}$ , X$_{2}$ ... be independent copies of the rv X in R d with df F . We make no further assumption on F . Conditioning on X$_{1}$ = y we obtain P ( X$_{N}$$_{(2)}$ ≤ x ) = ∞ ∑ j =2 P ( X$_{j}$ ≤ x ,N (2) = j ) = ∞ ∑ j =2 ∫ R d P ( X$_{j}$ ≤ x , X$_{2}$ ≤ y ,..., X$_{j}$$_{−}$$_{1}$ ≤ y , X$_{j}$ ̸≤ y ) F (d y ) = ∞ ∑ j =2 ∫ R d F ( y ) j − $^{2}$P ( X$_{j}$ ≤ x , X$_{j}$ ̸≤ y ) F (d y ) = ∫ R d P ( X ≤ x , X ̸≤ y ) ∞ ∑ j =2 F ( y ) j − $^{2}$F (d y ) = ∫ R d P ( X ≤ x , X ̸≤ y ) 1 − F ( y ) F (d y ) = R d P ( X ≤ x | X ̸≤ y ) F (d y ) . E ( N ( n + 1) − N ( n )) = ∫ ∞ 0 P ( N ( n + 1) − N ( n ) > t d t ) ≤ ∞ 0 P ( N ( n + 2) − N ( n + 1) > t d t ) = E ( N ( n + 2) − N ( n + 1)) . □ P ( N ( n + 1) − N ( n ) ≤ t ) = ∫ [0 , 1] d P ( N ( n + 1) − N ( n ) ≤ t | X$_{N}$$_{(}$$_{n}$$_{)}$ = x ) ( P ∗ X$_{N}$$_{(}$$_{n}$$_{)}$ )(d x ) = ∫ [0 , 1] d 1 − C ( x ) ⌊ t ⌋ ( P ∗ X$_{N}$$_{(}$$_{n}$$_{)}$ )(d x ) = 1 − E ( C ( X$_{N}$$_{(}$$_{n}$$_{)}$ ) ⌊ t ⌋ ) ,