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 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 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 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 P ( ω ′′ ˜ $_{M$_{n}$}$( δ ) ≥ ε ) ≤ η, n ≥ n$_{0}$, M ′′ m := max 0 ≤ i ≤ j ≤ k ≤ $_{m}$min {| S$_{j}$ − S$_{i}$ | , | S$_{k}$ − S$_{j}$ |} , P ( M ′′ m ≥ λ ) ≤ K λ $_{2}$( u$_{1}$ + . . . + u$_{m}$ ) (S1.5) P ( {| S$_{j}$ − S$_{i}$ | ≥ λ, | S$_{k}$ − S$_{j}$ | ≥ λ } ) ≤ ε$_{n}$ λ 2 ∑ i<l ≤ k u$_{l}$, 0 ≤ i ≤ j ≤ k ≤ m, ω ′′ $_{v}$( δ ) = sup min {| v ( x ) − v ( x$_{1}$ ) | , | v ( x$_{2}$ ) − v ( x ) |} , x$_{1}$ ≤ x ≤ x$_{2}$, x$_{2}$ − x$_{1}$ ≤ δ. 39