As a result, E ( α$_{i}$ ( δ$_{n}$, 0) α$_{j}$ ( δ$_{n}$, 0)) = 0. Moreover, As a result, Lemma S1.3 yields E( M ∗ $_{n}$( t )) = 0 and where $^{F}$⌊ nt ⌋ denotes the empirical distribution function of ˜ ψ ( X$_{i}$ ), i ≥ 1, i.e. $^{Note that sup$_{0}$}$≤ x ≤ 1 ( $^{F}$⌊ nt ⌋ ( δ$_{n}$ + x ) − $^{F}$⌊ nt ⌋ ( x ) ) converges to 0 almost surely due to the Glivenko-Cantelli theorem for stationary, ergodic sequences and since F$_{˜}$ ψ ( X 1 ) is a con- tinuous distribution function. It follows that its expected value converges to 0, as well. Therefore, the right-hand side of (S1.12) converges to 0. Thus, M ∗ $_{n}$( t ) converges to 0 in probability. This implies fidi-convergence of M ∗ n as a process with values in D [0 , 1]. Again, E ( α 2 $_{j}$( δ$_{n}$, 0) ) = E ( $^{1}${ 0 < ˜ ψ ( X$_{j}$ ) − S$_{n}$ ≤ δ$_{n}$ } ) − 2 E ( $^{1}${ 0 < ˜ ψ ( X$_{j}$ ) − S$_{n}$ ≤ δ$_{n}$ } E ( $^{1}${ 0 < ˜ ψ ( X$_{j}$ ) − S$_{n}$ ≤ δ$_{n}$ } | F j − 1 )) + E ( E ( $^{1}${ 0 < ˜ ψ ( X$_{j}$ ) − S$_{n}$ ≤ δ$_{n}$ } E $^{1}${ 0 < ˜ ψ ( X$_{j}$ ) − S$_{n}$ ≤ δ$_{n}$ } | F j − 1 ) | F j − 1 )) = E ( $^{1}${ 0 < ˜ ψ ( X$_{j}$ ) − S$_{n}$ ≤ δ$_{n}$ } ) − E ( $^{1}${ 0 < ˜ ψ ( X$_{j}$ ) − S$_{n}$ ≤ δ$_{n}$ } E ( $^{1}${ 0 < ˜ ψ ( X$_{j}$ ) − S$_{n}$ ≤ δ$_{n}$ } | F j − 1 )) ≤ E $^{1}${ 0 < ˜ ψ ( X$_{j}$ ) − S$_{n}$ ≤ δ$_{n}$ } = F$_{˜}$ ψ ( X$_{j}$ ) − $_{S$_{n}$}$( δ$_{n}$ ) − F$_{˜}$ ψ ( X$_{j}$ ) − $_{S$_{n}$}$(0) . Var   ⌊ nt ⌋ ∑ j = ⌊ ns ⌋ +1 α$_{j}$ ( δ$_{n}$, 0)   = ⌊ nt ⌋ ∑ j = ⌊ ns ⌋ +1 E ( α 2 $_{j}$( δ$_{n}$, 0) ) ≤ ⌊ nt ⌋ ∑ j = ⌊ ns ⌋ +1 ( F$_{˜}$ ψ ( X$_{j}$ ) − $_{S$_{n}$}$( δ$_{n}$ ) − F$_{˜}$ ψ ( X$_{j}$ ) − $_{S$_{n}$}$(0) ) . Var $_{M}$∗ $_{n}$( t ) ≤ t E ( sup 0 ≤ x ≤ 1 ( $^{F}$⌊ nt ⌋ ( δ$_{n}$ + x ) − $^{F}$⌊ nt ⌋ ( x ) ) ) , (S1.12) $^{F}$⌊ nt ⌋ ( x ) := 1 ⌊ nt ⌋ ⌊ nt ⌋ ∑ i =1 $^{1}${ ˜ ψ ( X$_{i}$ ) ≤ x } . 44