Lemma 10.2.1. It holds that
∫ 1 0
∑[nu]
j=1(Cjn−Zj)
√ℓ([nu])
2
du→p 0, where ℓ(n) = ∑n
j=11/(2j).
Proof of the Lemma 10.2.1. The left-hand side is evaluated by
∫ 1 0
∑n
j=11{j ≤nu}(Cjn−Zj)
√ℓ([nu])
2
du
=
∫ 1 0
n
∑
j=1 n
∑
k=1
1{j ≤nu}(Cjn−Zj)1{k ≤nu}(Ckn−Zk)
ℓ([nu]) du
=
n
∑
j=1 n
∑
k=1
∫ 1 0
1{j ∨k≤nu}(Cjn−Zj)(Ckn−Zk)
ℓ([nu]) du
≤
n
∑
j=1 n
∑
k=1
∫ 1 0
1{j ∨k≤nu}
ℓ([nu]) du|(Cjn−Zj)(Ckn−Zk)|
By the similar way as (9.2.6), the right-hand side is bounded above by 2
log(n) ( 1
log 2 + log log(n) )
|(C1n−Z1)(C1n−Z1)|
− 2 log log 2
log(n) |(C2n−Z2)((C1n−Z1) + (C2n+Z2))| +
n
∑
j=1 n
∑
k=2
2 log log(n)
log(n) |(Cjn−Zj)(Ckn−Zk)|
< 2 log(n)
( 1
log 2 −log log 2 + log log(n) )( n
∑
j=1
|Cjn−Zj| )2
So, it is sufficient to prove that
√log log(n) log(n)
n
∑
j=1
|Cjn−Zj| →p 0.
For any 1 ≤b=b(n)≤n, the triangle inequality yields that
n
∑
j=1
|Cjn−Zj| ≤
b
∑
j=1
|Cjn−Zj|+
n
∑
j=b+1
Cjn+
n
∑
j=b+1
Zj.
By the way similar to the equation (22) in Arratia et al. (1995), fix a good coupling such that the first term in the right-hand side is bounded by the total variation distance which converges to 0 and both of the expectation of the second term and third term are O(log log(n)) where we let b(n) =n/log(n).
This completes the proof.
This lemma yields the following corollary, which is a Poisson process ap-proximation.
Corollary 10.2.1. If Lemma 10.2.1 holds, then it holds that
∫ 1 0
∑[nu]
j=1(Cjn−Zj)
√ℓ′([nu])
2
du→p 0.
Proof of the Corollary 10.2.1. Consider random variables Pj ∼ P ois(λj =j) j = 1,2, . . . . By the definition of the median, it holds that
P(Pj < med(Pj))< 1 2. On the other hand, it holds that
P(Pj < j) =e−j
j−1
∑
i=0
ji
i! =jλj.
Teicher (1955) proves that jλj is increasing as j goes larger, which is stated in their second inequality of (8). The convergence (20) in Donnelly et al.
(1991a)
jλj → 1 2 yields that
1
2j > λj, ∀j = 1,2, . . . . (10.2.1) Since it holds that
∞
∑
j=1
( 1 2j −λj
)
= 1
2log(2),
which is the equation (31) in Donnelly et al. (1991a) and (10.2.1), it holds that
0< sup
u∈[0,1]
(ℓ([nu])−ℓ′([nu])) =
n
∑
j=1
( 1 2j −λj
)
=O(1)
and infu∈[0,1]ℓ′([nu]) = ℓ′(1) = 1/e. Thence, we have
∫ 1 0
∑[nu]
j=1(Cjn−Zj)
√ℓ′([nu])
2
du
=
∫ 1 0
∑[nu]
j=1(Cjn−Zj)
√ℓ([nu])
2(
1 + ℓ([nu])−ℓ′([nu]) ℓ′([nu])
) du
≤
∫ 1 0
∑[nu]
j=1(Cjn−Zj)
√ℓ([nu])
2(
1 + supu∈[0,1](ℓ([nu])−ℓ′([nu])) infu∈[0,1]ℓ′([nu])
) du
= (
1 + ℓ(n)−ℓ′(n) ℓ′(1)
) ∫ 1 0
∑[nu]
j=1(Cjn−Zj)
√ℓ([nu])
2
du
= (
1 +e
n
∑
j=1
( 1 2j −λj
))
∫ 1 0
∑[nu]
j=1(Cjn−Zj)
√ℓ([nu])
2
du→p 0.
This completes the proof.
This corollary and the functional CLT for a Poisson process yields the Theorem 10.1.1.
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Acknowledgements
The author would like to state his sincere thanks to those who gave com-ments to the contents and who supported him in various senses.
The author is a Research Fellow of Japan Society for the Promotion of Science. This work is partly supported by JSPS KAKENHI Grant Number 26-1487 (Grant-in-Aid for JSPS Fellows).