Page 152 - Contributed Paper Session (CPS) - Volume 1

CPS1216 Teppei O.
Condition [A2] holds if observation times are generated by mixing
processes as seen in the following example.

Example 3.1. Let { } ≥0 be an exponential α-mixing point process with

stationary increments for k = 1,2. Assume that [| | ] < ∞ for any q > 0 and

1

k = 1,2. Set , = inf{ ≥ 0; ≥ }.Then [A2] is satisfied with ≡ [ ]
1
(constants) by Rosenthal-type inequalities (Theorem 4 in [4]) and a similar
argument to the proof of Proposition 6 in [4]. Also, [B2] (defined later) is
⁄
satisfied if further −3 5+ → 0 for some ϵ > 0.

We study results related to consistency under Conditions [A1] and [A2]. Let

1
̃ = (̃ , … , ̃ ), ̃ = ⁄ ,∗ for 1 ≤ ≤ , ∑ = (∑ )0 ≤ ≤ , ∑ () =

,†
†

(, , ), () = (∑ ()) 0≤≤ , and (, ) = ([ ] √̃ ̃ ) for =

1≤,≤
( ) : × matrix valued.
≥0
We define
1 −1 2
⁄
(∑(), ∑ ) = ∫ { (((, ∑ ) − (, ∑()), (, ∑()) ) −
†
†
0
4
1 ⁄ 1 ⁄
((, ∑ ) 1 2 ) + ((, ∑()) 1 2 )}
2 † 2
2
⁄
⁄
⁄
= ∫ (((, ∑ ) 1 2 − (, ∑()) 1 2 ) (, ∑()) −1 2 ) .
†

Theorem 3.1. Assume [A1], [A2] and [V]. Then
((̂ ), ∑ ) → min(∑(), ∑ )

†
†

as → ∞.
−1 −1
Under boundedness of + ( ) and ∥Σ(σ) ∥, there exist positive

constants C1 and C2 such that

2
2
∫ |∑(, , ) − ∑ | ≤ (∑(), ∑ ) ≤ ∫ |∑(, , ) − ∑ | (3.3)

,†
2
†
1
,†

0 0

−1
−1

where C1 and C2 depend only the upperbounds of + ( ) and ∥Σ(σ) ∥.

The proof is left in the appendix. Then we can say that D is equivalent to L 2
norm.

Remark 3.1. In the specified setting of Ogihara [3] , () =
1
(∑(), ∑( ))holds for () in Section 2.2 of [3]. The presentation 2.8 of
∗
1
1 2
⁄
, () is obtained by calculating elements of (∑()) for = 2.
1

3.2 Optimal rate convergence
In this section, we study optimal rate convergence. Ogihara [3] showed
that local asymptotic normality holds for the specified model with nonrandom
⁄
diffusion coefficients, the optimal rate of convergence is equal to 1 4 for
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