Appendix
B. Proofs of Main Theorems
−Fadj,j(Xi(j ))−FX(j )(Xk(j ))
S1+S2. (B.4)
First, we examineS1. SinceFadj,j(x)is the estimated cumulative distribution function with 0≤Fadj,j(x)≤1, then for anyiandk, we have that
Fadj,j(Xi(j ))−Fadj,j(Xk(j ))≤1. (B.5) By the triangle inequality, we have that
F (Yi)−F (Y k)− |F (Yi)−F (Yk)| ≤F (Yi)−F (Yi)+F (Yk)−F (Yk). (B.6) Then by (B.6), we have that
P&
F (Yi)−F (Y k)− |F (Yi)−F (Yk)| > ξ
'
≤PF (Yi)−F (Yi)+F (Yk)−F (Yk)> ξ
≤P (
sup
t∈[0,τ]
F (t )−F (t )+ sup
t∈[0,τ]
F (t )−F (t )> ξ )
=P (
2 sup
t∈[0,τ]
F (t )−F (t )> ξ )
=P (
sup
t∈[0,τ]F (t )−F (t )>ξ 2
)
≤κ1exp
−1
4nξ2η4κ2
, (B.7)
where the last step is due to Lemma1withξin the right-hand side of (A.1) replaced byξ2. Therefore, combining (B.5) and (B.7) gives
PS1> ξ
≤κ1exp
−1
4nξ2η4κ2
. (B.8)
Next, we examineS2in a similar manner. Similar to (B.5), we have that
|F (Yi)−F (Yk)| ≤1. (B.9)
Similar to the arguments for (B.7), we obtain that P&
Fadj,j(Xi(j ))−FX(j )(Xk(j ))−Fadj,j(Xi(j ))−FX(j )(Xk(j )) > ξ
'
≤κ3exp
&
−nξ2
2G2 +o(n−15) '
. (B.10)
Therefore, combining (B.9) and (B.10) yields P S2> ξ
≤κ3exp
&
−nξ2
2G2+o(n−15) '
. (B.11)
Finally, combining (B.4), (B.8), and (B.11), the probabilistic bound ofMj,1− Mj,1is given by
P Mj,1−Mj,1> ξ
=PS1+S2> ξ
(B.12)
≤PS1+S2> ξ
≤PS1> ξ 2
+PS2>ξ 2
≤κ1exp
−1
16nξ2η4κ2
+κ3exp
&
−nξ2
8G2+o(n−15) '
.
Furthermore, similar derivations show that P Mj,2−Mj,2> ξ
≤κ1exp
−1
16nξ2η4κ2
+κ3exp
&
−nξ2
8G2+o(n−15) '
(B.13) and
P Mj,3−Mj,3> ξ
≤κ1exp
−1
16nξ2η4κ2
+κ3exp
&
−nξ2
8G2 +o(n−15) '
. (B.14)
Noting that the upper bounds in (B.12)–(B.14) are dominated by exp
−c∗nξ2 for certain constantc∗, we apply (B.12)–(B.14) to (B.3) and obtain that
Pωj−ωj∗> ξ =O
exp
−c2nξ2
(B.15)
for somec2 > 0. Thus, combining (B.2) and (B.15) with (B.1) and specifying ξ =cn−ζ for the constantscandζ described in Condition (C4) yield the desired result.
Part 2 We prove(20).
LetJ =min
j∈Iωj−max
j∈Icωj. The left-hand side of (20) can be expressed as P
max
j∈Icωj≥min j∈Iωj
=P
max
j∈Icωj−max
j∈Icωj≥min
j∈Iωj−max j∈Icωj
=P
max
j∈Icωj−max
j∈Icωj≥min
j∈Iωj−min j∈Iωj+J
≤P
max
j∈Icωj−ωj+max
j∈Iωj−ωj≥J
≤P
2 max
j=1,···,pωj−ωj≥J
=O
&
exp
−1 4Dnv20
' ,
where the last step comes from the result in Part 1 and Condition (C5).
B.2 Proof of Theorem2
Similar to the derivations of Li et al. (2012), one can obtain that
&
maxj∈Iωj−ωj≤cn−ζ '
⊆ I⊆I
. It gives
P I⊆I
≥1−P
max
j∈Iωj−ωj≤cn−ζ
≥1−qPωj−ωj≤cn−ζ
≥1−O
qexp
−Dn1−2ζ
,
where the last step comes from Theorem1.
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