S1 Proof of Lemma 3.6 Assume that { X$_{j}$, j ≥ 1 } follows the LMSV model. Define Lemma 3.6 states that in D ([ −∞ , ∞ ] × [0 , 1]) Proof of Lemma 3.6. In order to prove Lemma 3.6 we have to verify tightness in D ([ −∞ , ∞ ] × [0 , 1]). For this we quote Theorem 3.1 in Ivanoff (1983). Theorem S1.1. Let ( X$_{n}$, F $_{n}$, P$_{n}$ ) be a sequence of probability spaces such that X$_{n}$ is a random element of D ([0 , 1] × [0 , 1]) for each n , and the process X$_{n}$ ( x, t ) is adapted to a complete, right continuous filtration F $_{n}$( x, t ) ⊆ F $_{n}$. If X$_{n}$ ( x, t ) is tight for each ( x, t ) ∈ [0 , 1] × [0 , 1] and if for all δ$_{n}$ ↘ 0 and each canonical 1-stopping time S$_{n}$ and each canonical 2-stopping time T$_{n}$ then X$_{n}$ is tight in D ([0 , 1] × [0 , 1]) . In the following we will show that the conditions of the theorem hold for X$_{n}$ ( x, t ) := n − 1 / $^{2}$M$_{n}$ ◦ H , where H : [0 , 1] × [0 , 1] −→ [ −∞ , ∞ ] × [0 , 1] with H ( x, t ) = ( h ( x ) , t ) for some increasing isomorphism h : [0 , 1] −→ [ −∞ , ∞ ]. Then, Lemma 3.6 immediately follows from an application of Theorem S1.1. Indeed, due to Lemma 3.5, n − 1 / $^{2}$M$_{n}$ ◦ H ( x, t ) is tight for each ( x, t ) ∈ [0 , 1] × [0 , 1]. 35 M$_{n}$ ( x, t ) := ⌊ nt ⌋ ∑ j =1 ( $^{1}${ ψ ( X$_{j}$ ) ≤ x } − E ( $^{1}${ ψ ( X$_{j}$ ) ≤ x } | F j − 1 )) . 1 √$_{n}$ M$_{n}$ ( x, t ) = O $_{P}$(1) sup 0 ≤ t ≤ $_{1}$| X$_{n}$ ( S$_{n}$ + δ$_{n}$, t ) − X$_{n}$ ( S$_{n}$, t ) | P −→ 0 , as n → ∞ , (S1.1) sup 0 ≤ x ≤ $_{1}$| X$_{n}$ ( x, T$_{n}$ + δ$_{n}$ ) − X$_{n}$ ( x, T$_{n}$ ) | P −→ 0 , as n → ∞ , (S1.2)