content_image/296180.jpg content_image/296181.jpg content_image/296182.jpg content_image/296183.jpg content_image/296184.jpg
with 0 ≤ x$_{1}$ ≤ . . . ≤ x$_{r}$ ≤ 1 and x$_{k}$ − $^{x$_{k}$}$− 1 ≤ 2 δ . Hence, we may conclude that 

<img src='content_image/296180.jpg'>

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 

<img src='content_image/296181.jpg'>

<img src='content_image/296182.jpg'>

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 

<img src='content_image/296183.jpg'>

<img src='content_image/296184.jpg'>

