As a result, It follows that Due to Lemma S1.2 E( ˜ M$_{n}$ ( x )) = 0 and Hence, ˜ M$_{n}$ ( x ) converges to 0 in probability. This implies fidi-convergence of ˜ M$_{n}$ as a process with values in D [0 , 1]. 38 Var( ˜ M$_{n}$ ( x )) ≤ 1 n ⌊ nδ$_{n}$ ⌋ ∑ i =1 F$_{˜}$ ψ ( X$_{i}$$_{+}$$_{τ}$$_{n}$ $_{)}$( x ) ≤ δ$_{n}$ −→ 0 . Var  $^{τ$_{n}$}$$^{+}$$^{⌊}$$^{nδ$_{n}$}$$^{⌋}$ ∑ j = τ$_{n}$ +1 α$_{j}$ ( y, x )   = E    $^{τ$_{n}$}$$^{+}$$^{⌊}$$^{nδ$_{n}$}$$^{⌋}$ ∑ j = τ$_{n}$ +1 α$_{j}$ ( y, x )  $^{2}$   ≤ ⌊ nδ$_{n}$ ⌋ ∑ j =1 [ F$_{˜}$ ψ ( X$_{j}$$_{+}$$_{τ}$$_{n}$ $_{)}$( y ) − F$_{˜}$ ψ ( X$_{j}$$_{+}$$_{τ}$$_{n}$ $_{)}$( x ) ] . Moreover, we have E ( α$_{i}$$_{+}$$_{k}$ ( y, x ) α$_{j}$$_{+}$$_{k}$ ( y, x $^{)1}${ τ$_{n}$ = k } ) = 0 . n ∑ k =1 E ( α 2 i + $_{k}$( y, x $^{)1}${ τ$_{n}$ = k } ) = n ∑ k =1 E [ ( $^{1}${ x< ˜ ψ ( X$_{i}$$_{+}$$_{k}$ ) ≤ y } − E ( $^{1}${ x< ˜ ψ ( X$_{i}$$_{+}$$_{k}$ ) ≤ y } | F i + k − 1 )) $^{$^{2}$1}${ τ$_{n}$ = k } ] = n ∑ k =1 E [ ( $^{1}${ x< ˜ ψ ( X$_{i}$$_{+}$$_{k}$ ) ≤ y } $^{1}${ τ$_{n}$ = k } − $^{21}${ x< ˜ ψ ( X$_{i}$$_{+}$$_{k}$ ) ≤ y } E ( $^{1}${ x< ˜ ψ ( X$_{i}$$_{+}$$_{k}$ ) ≤ y } $^{1}${ τ$_{n}$ = k } | F i + k − 1 )) + E ( $^{1}${ x< ˜ ψ ( X$_{i}$$_{+}$$_{k}$ ) ≤ y } E ( $^{1}${ x< ˜ ψ ( X$_{i}$$_{+}$$_{k}$ ) ≤ y } $^{1}${ τ$_{n}$ = k } | F i + k − 1 ) | F i + k − 1 ) ] = n ∑ k =1 { E ( $^{1}${ x< ˜ ψ ( X$_{i}$$_{+}$$_{k}$ ) ≤ y } $^{1}${ τ$_{n}$ = k } ) − E ( $^{1}${ x< ˜ ψ ( X$_{i}$$_{+}$$_{k}$ ) ≤ y } E ( $^{1}${ x< ˜ ψ ( X$_{i}$$_{+}$$_{k}$ ) ≤ y } $^{1}${ τ$_{n}$ = k } | F i + k − 1 ))} ≤ n ∑ k =1 E [ $^{1}${ x< ˜ ψ ( X$_{i}$$_{+}$$_{k}$ ) ≤ y } $^{1}${ τ$_{n}$ = k } ] = n ∑ k =1 E [ $^{1}${ x< ˜ ψ ( X$_{i}$$_{+}$$_{τ}$$_{n}$ ) ≤ y } $^{1}${ τ$_{n}$ = k } ] = E [ $^{1}${ x< ˜ ψ ( X$_{i}$$_{+}$$_{τ}$$_{n}$ ) ≤ y } ] = F$_{˜}$ ψ ( X$_{i}$$_{+}$$_{τ}$$_{n}$ $_{)}$( y ) − F$_{˜}$ ψ ( X$_{i}$$_{+}$$_{τ}$$_{n}$ $_{)}$( x ) .