Recall that F j = σ ( ε$_{j}$$^{, ε$_{j}$}$− $_{1}$, . . . , η$_{j}$ $^{, η$_{j}$}$− $_{1}$, . . . ). Define for all x ∈ [0 , 1]. Then, X$_{n}$ ( x, t ) is adapted to F $_{n}$( x, t ). Moreover, the corresponding filtration is right-continuous. Let T$_{n}$ denote a canonical 2-stopping time for X$_{n}$ ( x, t ). In particular, we require { T$_{n}$ ≤ t } to be measurable with respect to F $_{n}$(1 , t ) = F ⌊ nt ⌋ . Define τ$_{n}$ := ⌊ nT$_{n}$ ⌋ . Note that with For 0 ≤ x ≤ 1 define In order to show (S1.2), we have to prove that Prior to the proof, we establish the following result: Lemma S1.2. We have E [ ˜ M$_{n}$ ( x ) ] = 0 for all x ∈ R . For x < y F $_{n}$( x, t ) := F ⌊ nt ⌋ ∣ 1 √$_{n}$ M$_{n}$ ( y, T$_{n}$ + δ$_{n}$ ) − 1 √$_{n}$ M$_{n}$ ( y, T$_{n}$ ) ∣ = ∣ 1 √$_{n}$ τ$_{n}$ + ⌊ nδ$_{n}$ ⌋ ∑ j = τ$_{n}$ +1 ζ$_{j}$ ( y ) ∣ ζ$_{j}$ ( y $^{) = 1}${ ψ ( X$_{j}$ ) ≤ y } − E ( $^{1}${ ψ ( X$_{j}$ ) ≤ y } | F j − 1 ) . ˜ M$_{n}$ ( x ) := 1 √$_{n}$ τ$_{n}$ + ⌊ nδ$_{n}$ ⌋ ∑ j = τ$_{n}$ +1 ζ$_{j}$ ( h ( x )) . sup 0 ≤ x ≤ 1 ∣ ˜ M$_{n}$ ( x ) ∣ P −→ 0 , as n → ∞ . (S1.3) Var ( ˜ M$_{n}$ ( y ) − ˜ M$_{n}$ ( x ) ) = E [ ( ˜ M$_{n}$ ( y ) − ˜ M$_{n}$ ( x ) ) 2 ] ≤ 1 n ⌊ nδ$_{n}$ ⌋ ∑ j =1 [ F$_{˜}$ ψ ( X$_{j}$$_{+}$$_{τ}$$_{n}$ $_{)}$( y ) − F$_{˜}$ ψ ( X$_{j}$$_{+}$$_{τ}$$_{n}$ $_{)}$( x ) ] , 36