we make use of Theorem 15.4 in Billingsley (1968), i.e. we verify that for each positive ε and η , there exists a δ , 0 < δ < 1, and an integer n$_{0}$ such that Define where S$_{i}$ = M ∗ $_{n}$( τ + i $_{m}$δ ). For t$_{1}$ ≤ t ≤ t$_{2}$ we have The Cauchy-Schwarz inequality yields By the same argument as in the proof of (S1.2) it follows that for any sequence γ$_{n}$ , n ∈ N , converging to 0. Taking the right-continuity of M ∗ n into 45 P ( M ′′ m ≥ ε ) ≤ K ε 2 γ − 1 n n ⌊ n ( τ + δ ) ⌋ ∑ j = ⌊ nτ ⌋ +1 ( F$_{˜}$ ψ ( X$_{j}$ ) − $_{S$_{n}$}$( δ$_{n}$ ) − F$_{˜}$ ψ ( X$_{j}$ ) − $_{S$_{n}$}$(0) ) (S1.13) E | $_{M}$∗ $_{n}$( t ) − $_{M}$∗ $_{n}$( t$_{1}$ ) | | $_{M}$∗ $_{n}$( t$_{2}$ ) − $_{M}$∗ $_{n}$( t ) | ≤ √ E  $^{1}$ n   ⌊ nt ⌋ ∑ j = ⌊ nt 1 ⌋ +1 α$_{j}$ ( δ$_{n}$, 0)  $^{2}$   E  $^{1}$ n   ⌊ nt 2 ⌋ ∑ j = ⌊ nt ⌋ +1 α$_{j}$ ( δ$_{n}$, 0)  $^{2}$   ≤ 1 n ⌊ nt 2 ⌋ ∑ i = ⌊ nt 1 ⌋ +1 ( F$_{˜}$ ψ ( X$_{i}$ ) − $_{S$_{n}$}$( δ$_{n}$ ) − F$_{˜}$ ψ ( X$_{i}$ ) − $_{S$_{n}$}$(0) ) . M ′′ m := max 0 ≤ i ≤ j ≤ k ≤ $_{m}$min {| S$_{j}$ − S$_{i}$ | , | S$_{k}$ − S$_{j}$ |} , P ( ω ′′ M ∗ $_{n}$( δ ) ≥ ε ) ≤ η, n ≥ n$_{0}$. | $_{M}$∗ $_{n}$( t ) − $_{M}$∗ $_{n}$( t$_{1}$ ) | = 1 √$_{n}$ ∣ ⌊ nt ⌋ ∑ j = ⌊ nt 1 ⌋ +1 α$_{j}$ ( δ$_{n}$, 0) ∣ ∣$^{,}$ | $_{M}$∗ $_{n}$( t$_{2}$ ) − $_{M}$∗ $_{n}$( t ) | = 1 √$_{n}$ ∣ ⌊ nt 2 ⌋ ∑ j = ⌊ nt ⌋ +1 α$_{j}$ ( δ$_{n}$, 0) ∣