where F$_{˜}$ ψ ( X$_{j}$$_{+}$$_{τ}$$_{n}$ $_{)}$( x ) := P ( ˜ ψ ( X$_{j}$$_{+}$$_{τ}$$_{n}$ ) ≤ x ) and ˜ ψ := h − 1 ◦ ψ . Proof. Define Then Since { τ$_{n}$ = k } is measurable with respect to F j for all j ≥ k , it follows that where ˜ ψ := h − 1 ◦ ψ . Furthermore, For i < j α$_{j}$ ( y, x $^{) := 1}${ h ( x ) <ψ ( X$_{j}$ ) ≤ h ( y ) } − E ( $^{1}${ h ( x ) <ψ ( X$_{j}$ ) ≤ h ( y ) } | F j − 1 ) . (S1.4) ˜ M$_{n}$ ( y ) − ˜ M$_{n}$ ( x ) = 1 √$_{n}$ τ$_{n}$ + ⌊ nδ$_{n}$ ⌋ ∑ j = τ$_{n}$ +1 α$_{j}$ ( y, x ) . E  $^{τ$_{n}$}$$^{+}$$^{⌊}$$^{nδ$_{n}$}$$^{⌋}$ ∑ j = τ$_{n}$ +1 α$_{j}$ ( y, x )   = n ∑ k =1 ⌊ nδ$_{n}$ ⌋ ∑ j =1 E ( α$_{j}$$_{+}$$_{k}$ ( y, x $^{)1}${ τ$_{n}$ = k } ) = n ∑ k =1 ⌊ nδ$_{n}$ ⌋ ∑ j =1 E ( $^{1}${ x< ˜ ψ ( X$_{j}$$_{+}$$_{k}$ ) ≤ y } $^{1}${ τ$_{n}$ = k } − E ( $^{1}${ x< ˜ ψ ( X$_{j}$$_{+}$$_{k}$ ) ≤ y } $^{1}${ τ$_{n}$ = k } | F j + k − 1 )) = 0 , E    $^{τ$_{n}$}$$^{+}$$^{⌊}$$^{nδ$_{n}$}$$^{⌋}$ ∑ j = τ$_{n}$ +1 α$_{j}$ ( y, x )  $^{2}$   = n ∑ k =1 ⌊ nδ$_{n}$ ⌋ ∑ i =1 ⌊ nδ$_{n}$ ⌋ ∑ j =1 E ( α$_{i}$$_{+}$$_{k}$ ( y, x ) α$_{j}$$_{+}$$_{k}$ ( y, x $^{)1}${ τ$_{n}$ = k } ) . α$_{i}$$_{+}$$_{k}$ ( y, x ) α$_{j}$$_{+}$$_{k}$ ( y, x $^{)1}${ τ$_{n}$ = k } $^{= 1}${ x< ˜ ψ ( X$_{i}$$_{+}$$_{k}$ ) ≤ y } $^{1}${ x< ˜ ψ ( X$_{j}$$_{+}$$_{k}$ ) ≤ y } $^{1}${ τ$_{n}$ = k } + E ( $^{1}${ x< ˜ ψ ( X$_{j}$$_{+}$$_{k}$ ) ≤ y } $^{1}${ τ$_{n}$ = k } E ( $^{1}${ x< ˜ ψ ( X$_{i}$$_{+}$$_{k}$ ) ≤ y } | F i + k − 1 ) | F j + k − 1 ) − E $^{1}${ x< ˜ ψ ( X$_{i}$$_{+}$$_{k}$ ) ≤ y } $^{1}${ x< ˜ ψ ( X$_{j}$$_{+}$$_{k}$ ) ≤ y } $^{1}${ τ$_{n}$ = k } | F j + k − 1 ) − $^{1}${ x< ˜ ψ ( X$_{j}$$_{+}$$_{k}$ ) ≤ y } $^{1}${ τ$_{n}$ = k } E ( $^{1}${ x< ˜ ψ ( X$_{i}$$_{+}$$_{k}$ ) ≤ y } | F i + k − 1 . 37