with 0 ≤ x$_{1}$ ≤ . . . ≤ x$_{r}$ ≤ 1 and x$_{k}$ − $^{x$_{k}$}$− 1 ≤ 2 δ . Hence, we may conclude that Since the right-hand side of the above inequality converges to 0, (S1.2) has been proved. Proving (S1.1) . It remains to show (S1.1). For this, let S$_{n}$ denote a canonical 1-stopping time for X$_{n}$ ( x, t ). Note that with where ˜ ψ := h − 1 ◦ ψ . Prior to the proof of (S1.1), we establish the following result: Lemma S1.3. We have E( M ∗ $_{n}$( t )) = 0 for all t ∈ [0 , 1] . For s < t where F$_{˜}$ ψ ( X$_{i}$ ) − $_{S$_{n}$}$( x ) := P ( ˜ ψ ( X$_{i}$ ) − S$_{n}$ ≤ x ) . Proof. Note that P ( $_{ω}$′′$_{˜}$ $_{M$_{n}$}$( δ ) ≥ ε ) ≤ 2 K ε 2 δ 1 − γ n . sup 0 ≤ t ≤ 1 | X$_{n}$ ( S$_{n}$ + δ$_{n}$, t ) − X$_{n}$ ( S$_{n}$, t ) | = sup 0 ≤ t ≤ 1 | $_{M}$∗ $_{n}$( t ) | $_{M}$∗ $_{n}$( t ) := 1 √$_{n}$ ⌊ nt ⌋ ∑ j =1 ( $^{1}${ h ( S$_{n}$ ) <ψ ( X$_{j}$ ) ≤ h ( S$_{n}$ + δ$_{n}$ ) } − E ( $^{1}${ h ( S$_{n}$ ) <ψ ( X$_{j}$ ) ≤ h ( S$_{n}$ + δ$_{n}$ ) } | F j − 1 )) = 1 √$_{n}$ ⌊ nt ⌋ ∑ j =1 ( $^{1}${ 0 < ˜ ψ ( X$_{j}$ ) − S$_{n}$ ≤ δ$_{n}$ } − E ( $^{1}${ 0 < ˜ ψ ( X$_{j}$ ) − S$_{n}$ ≤ δ$_{n}$ } | F j − 1 )) , Var( $_{M}$∗ $_{n}$( t ) − $_{M}$∗ $_{n}$( s )) = E [ ( $_{M}$∗ $_{n}$( t ) − $_{M}$∗ $_{n}$( s )) $^{2}$] ≤ 1 n ⌊ nt ⌋ ∑ i = ⌊ ns ⌋ +1 ( F$_{˜}$ ψ ( X$_{i}$ ) − $_{S$_{n}$}$( δ$_{n}$ ) − F$_{˜}$ ψ ( X$_{i}$ ) − $_{S$_{n}$}$(0) ) , $_{M}$∗ $_{n}$( t ) − $_{M}$∗ $_{n}$( s ) = 1 √$_{n}$ ⌊ nt ⌋ ∑ j = ⌊ ns ⌋ +1 α$_{j}$ ( δ$_{n}$, 0) , 42