ON FUNCTIONAL RECORDS AND CHAMPIONS 17 where ξ is a rv with P ( ξ = 1) = γ = 1 − P ( ξ = 0), and ξ is independent of Z ∗ which is a random permutation of the vector ( d, 0 , . . . , 0) with equal probability 1 /d . Obviously, P ( Z > 0 , ξ = 0) = 0. On the other hand, ξ = 1 implies Z = 1. Thus, we obtain by (13) for all x ≤ 0 In order to generalize Proposition 3.7 to stochastic processes in C ( S ) with arbi- trary margins, the following lemma is needed. □ Lemma 3.9. Let f$_{n}$ , n ∈ N , be a sequence of functions in ¯ E $^{−}$( S ) converging uniformly to f ∈ ¯ E $^{−}$( S ) . Then, under the conditions and notation of Proposition 3.7, Proof. Let ε > 0. Due to the uniform convergence of f$_{n}$ , there exists N ∈ N such that f − ε ≤ f$_{n}$ ≤ f + ε for n ≥ N . Assume without loss of generality f + ε < 0, otherwise consider min( f + ε, 0). Clearly, for such n , Now with n → ∞ , Proposition 3.7 shows Now check and let ε ↓ 0. The assertion now follows from the dominated convergence theorem. □ inf s ∈ S | max ( η$_{s}$, f ( s ) ± ε ) | Z$_{s}$ ≤ − η$_{s}$$_{0}$Z$_{s}$$_{0}$, s$_{0}$ ∈ S, E ( inf s ∈ S | max ( η$_{s}$, f ( s ) − ε ) | Z$_{s}$ ) ≤ lim n $_{→∞}$Π$_{n}$ ( f$_{n}$ ) ≤ E ( inf s ∈ S | max ( η$_{s}$, f ( s ) + ε ) | Z$_{s}$ ) . Π$_{n}$( f + ε ) ≤ Π$_{n}$( f$_{n}$ ) ≤ Π$_{n}$( f − ε ) . ¯ H$_{n}$ ( f$_{n}$ ) = Π$_{n}$( f$_{n}$ ) π$_{n}$ →$_{n}$$_{→∞}$ ¯ H$_{D}$ ( f ) . ¯ H$_{γ}$ ( x ) = 1 − E ( ( ‖ 1 / Z ‖$_{M}$$_{γ}$ ) − $^{1}$exp ( ‖ 1 / Z ‖$_{M}$$_{γ}$ max i =1 $_{,...,d}$( x$_{i}$Z$_{i}$ ) · 1$_{{}$ Z > 0 ,ξ =1 } ) E ( ‖ 1 / Z ‖$_{M}$$_{γ}$ ) − 1 · 1$_{{}$ Z > 0 ,ξ =1 } ) = 1 − exp ( ‖ 1 ‖$_{M}$$_{γ}$ max i =1 $_{,...,d}$x$_{i}$ ) .