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## 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]) 

<img src='content_image/298986.jpg'>

<img src='content_image/298987.jpg'>

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$_{n}$ 

<img src='content_image/298988.jpg'>

<img src='content_image/298989.jpg'>

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]. 

