E [∣ ∣ ˜ M$_{n}$ ( x$_{2}$ ) − ˜ M$_{n}$ ( x ) ∣ ∣ ˜ M$_{n}$ ( x ) − ˜ M$_{n}$ ( x$_{1}$ ) ∣ ] = E   1 √$_{n}$ ∣ ∣$^{τ$_{n}$}$$^{+}$$^{⌊}$$^{nδ$_{n}$}$$^{⌋}$ ∑ j = τ$_{n}$ +1 α$_{j}$ ( x$_{2}$, x ) ∣ 1 √$_{n}$ ∣ ∣$^{τ$_{n}$}$$^{+}$$^{⌊}$$^{nδ$_{n}$}$$^{⌋}$ ∑ j = τ$_{n}$ +1 α$_{j}$ ( x, x$_{1}$ ) ∣   ≤ √ √$^{1}$ n E    $^{τ$_{n}$}$$^{+}$$^{⌊}$$^{nδ$_{n}$}$$^{⌋}$ ∑ j = τ$_{n}$ +1 α$_{j}$ ( x$_{2}$, x )  $^{2}$   1 n E    $^{τ$_{n}$}$$^{+}$$^{⌊}$$^{nδ$_{n}$}$$^{⌋}$ ∑ j = τ$_{n}$ +1 α$_{j}$ ( x, x$_{1}$ )  $^{2}$   ≤ 1 $_{n}$√ √ √$^{⌊}$$^{nδ$_{n}$}$$^{⌋}$ ∑ i =1 ( F$_{˜}$ ψ ( X$_{i}$$_{+}$$_{τ}$$_{n}$ $_{)}$( x ) − F$_{˜}$ ψ ( X$_{i}$$_{+}$$_{τ}$$_{n}$ $_{)}$( x$_{1}$ ) ) √ √$^{⌊}$$^{nδ$_{n}$}$$^{⌋}$ ∑ i =1 ( F$_{˜}$ ψ ( X$_{i}$$_{+}$$_{τ}$$_{n}$ $_{)}$( x$_{2}$ ) − F$_{˜}$ ψ ( X$_{i}$$_{+}$$_{τ}$$_{n}$ $_{)}$( x ) ) ≤ 1 n ⌊ nδ$_{n}$ ⌋ ∑ i =1 ( F$_{˜}$ ψ ( X$_{i}$$_{+}$$_{τ}$$_{n}$ $_{)}$( x$_{2}$ ) − F$_{˜}$ ψ ( X$_{i}$$_{+}$$_{τ}$$_{n}$ $_{)}$( x$_{1}$ ) ) . (S1.6) The Markov inequality yields Therefore, it follows by (S1.6) that for some γ ∈ (0 , 1) where As a result, we get P ( | S$_{j}$ − S$_{i}$ | ≥ λ, | S$_{k}$ − S$_{j}$ | ≥ λ ) ≤ P ( | S$_{j}$ − S$_{i}$ | | S$_{k}$ − S$_{j}$ | ≥ λ $^{2}$) ≤ 1 λ 2 E | S$_{j}$ − S$_{i}$ | | S$_{k}$ − S$_{j}$ | . P ( | S$_{j}$ − S$_{i}$ | ≥ λ, | S$_{k}$ − S$_{j}$ | ≥ λ ) ≤ δ γ n λ 2 ∑ i<l ≤ k u$_{l}$, u$_{l}$ := δ − γ n n ⌊ nδ$_{n}$ ⌋ ∑ j =1 ( F$_{˜}$ ψ ( X$_{j}$$_{+}$$_{τ}$$_{n}$ ) ( τ + l m δ ) − F$_{˜}$ ψ ( X$_{j}$$_{+}$$_{τ}$$_{n}$ ) ( τ + l − 1 m δ )) . P ( M ′′ m ≥ ε ) ≤ K ε 2 δ − γ n n ⌊ nδ$_{n}$ ⌋ ∑ j =1 ( F$_{˜}$ ψ ( X$_{j}$$_{+}$$_{τ}$$_{n}$ $_{)}$( τ + δ ) − F$_{˜}$ ψ ( X$_{j}$$_{+}$$_{τ}$$_{n}$ $_{)}$( τ ) ) . (S1.7) 40