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Proving (S1.2) . In order to establish (S1.2), it remains to show tightness of ˜ M$_{n}$ . For this, we adopt the argument that proves Theorem 15.6 in Billingsley (1968). For any function v in D [0 , 1] define the modulus ω ′′ $_{v}$( δ ) by 

where the supremum extends over x$_{1}$ , x , and x$_{2}$ satisfying 

<img src='content_image/311635.jpg'>

<img src='content_image/311636.jpg'>

Under the assumption of fidi-convergence it suffices to show that for each positive ε and η , there exists a δ , 0 < δ < 1, and an integer n$_{0}$ such that 

(see Theorem 15.4 in Billingsley (1968)). 

Define 

<img src='content_image/311632.jpg'>

<img src='content_image/311631.jpg'>

where S$_{i}$ = ˜ M$_{n}$ ( τ + i $_{m}$δ ). Following the proof of Theorem 12.5 in Billingsley (1968), there exists an n$_{0}$ ∈ N such that 

holds for all positive λ , some constant K and n ≥ n$_{0}$ , if 

<img src='content_image/311634.jpg'>

<img src='content_image/311633.jpg'>

for some sequence ε$_{n}$ , n ∈ N , converging to 0. For x$_{1}$ ≤ x ≤ x$_{2}$ it follows by the Cauchy - Schwarz inequality for expected values and Lemma S1.2 that 

