content_image/318108.jpg content_image/318107.jpg content_image/318105.jpg content_image/318104.jpg content_image/318106.jpg
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 

<img src='content_image/318108.jpg'>

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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, 

<img src='content_image/318107.jpg'>

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 

<img src='content_image/318105.jpg'>

Now check 

<img src='content_image/318104.jpg'>

<img src='content_image/318106.jpg'>

and let ε ↓ 0. The assertion now follows from the dominated convergence theorem. 

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