where Obviously, Furthermore, The canonical 1-stopping time S$_{n}$ takes on only countably many values s$_{i}$ , i ∈ N ; see Ivanoff (1983). Note that for i < j α$_{j}$ ( y, x $^{) := 1}${ x< ˜ ψ ( X$_{j}$ ) − S$_{n}$ ≤ y } − E ( $^{1}${ x< ˜ ψ ( X$_{j}$ ) − S$_{n}$ ≤ y } | F j − 1 ) . (S1.11) E   ⌊ nt ⌋ ∑ j = ⌊ ns ⌋ +1 α$_{j}$ ( δ$_{n}$, 0)   = 0 . Var   ⌊ nt ⌋ ∑ j = ⌊ ns ⌋ +1 α$_{j}$ ( δ$_{n}$, 0)   = E     ⌊ nt ⌋ ∑ j = ⌊ ns ⌋ +1 α$_{j}$ ( δ$_{n}$, 0)  $^{2}$   = ⌊ nt ⌋ ∑ i = ⌊ ns ⌋ +1 ⌊ nt ⌋ ∑ j = ⌊ ns ⌋ +1 E ( α$_{i}$ ( δ$_{n}$, 0) α$_{j}$ ( δ$_{n}$, 0)) . α$_{i}$ ( δ$_{n}$, 0) α$_{j}$ ( δ$_{n}$, 0) = ∞ ∑ k =1 α$_{i}$ ( δ$_{n}$, 0) α$_{j}$ ( δ$_{n}$, $^{0)1}${ S$_{n}$ = s$_{k}$ } = ∞ ∑ k =1 $^{1}${ S$_{n}$ = s$_{k}$ } { $^{1}${ 0 < ˜ ψ ( X$_{i}$ ) − s$_{k}$ ≤ δ$_{n}$ } $^{1}${ 0 < ˜ ψ ( X$_{j}$ ) − s$_{k}$ ≤ δ$_{n}$ } + E ( $^{1}${ 0 < ˜ ψ ( X$_{j}$ ) − s$_{k}$ ≤ δ$_{n}$ } E ( $^{1}${ 0 < ˜ ψ ( X$_{i}$ ) − s$_{k}$ ≤ δ$_{n}$ } | F i − 1 ) | F j − 1 ) − E $^{1}${ 0 < ˜ ψ ( X$_{i}$ ) − s$_{k}$ ≤ δ$_{n}$ } $^{1}${ 0 < ˜ ψ ( X$_{j}$ ) − s$_{k}$ ≤ δ$_{n}$ } | F j − 1 ) − $^{1}${ 0 < ˜ ψ ( X$_{j}$ ) − s$_{k}$ ≤ δ$_{n}$ } E ( $^{1}${ 0 < ˜ ψ ( X$_{i}$ ) − s$_{k}$ ≤ δ$_{n}$ } | F i − 1 } $^{= 1}${ 0 < ˜ ψ ( X$_{i}$ ) − S$_{n}$ ≤ δ$_{n}$ } $^{1}${ 0 < ˜ ψ ( X$_{j}$ ) − S$_{n}$ ≤ δ$_{n}$ } + E ( $^{1}${ 0 < ˜ ψ ( X$_{j}$ ) − S$_{n}$ ≤ δ$_{n}$ } E ( $^{1}${ 0 < ˜ ψ ( X$_{i}$ ) − S$_{n}$ ≤ δ$_{n}$ } | F i − 1 ) | F j − 1 ) − E $^{1}${ 0 < ˜ ψ ( X$_{i}$ ) − S$_{n}$ ≤ δ$_{n}$ } $^{1}${ 0 < ˜ ψ ( X$_{j}$ ) − S$_{n}$ ≤ δ$_{n}$ } | F j − 1 ) − $^{1}${ 0 < ˜ ψ ( X$_{j}$ ) − S$_{n}$ ≤ δ$_{n}$ } E ( $^{1}${ 0 < ˜ ψ ( X$_{i}$ ) − S$_{n}$ ≤ δ$_{n}$ } | F i − 1 . 43