Page 151 - Contributed Paper Session (CPS) - Volume 1

CPS1216 Teppei O.
5. [min #{; , ∈ [ , )} ≥ 1] → 1 as → ∞ and {ℓn
1≤≤ −1
2
−1
maxm,k(#{j;Tj ∈ [sm−1,sm)} [|∑ , , | ]} is tight.
∏
;
∈[ −1 , )

() and
6. There exist progressively measurable processes { }

0≤≤,0≤≤1

()
()
{ ̂ () } such that sup [| | ⋁| ̂ | ] < ∞,

0≤≤,0≤≤1

for 0 ≤ j ≤ 1 and any q > 0, and

(0) (1) (1) (1) (0) (1)
,† = 0,† + ∫ + ∫ , = 0 + ∫ ̂ + ∫ ̂
0 0 0 0
for ∈ [0, ].
Some of these conditions are standard conditions in this field and easy to
check. Related to market microstructure noise , for X, point 5 of (A1) is
required. Roughly speaking, this condition is satisfied if the summation of
, is of an order equivalent to the square root of the number of , . This is

satisfied if sampling frequency of { , } is of order and , satisfies certain

independency, martingale conditions or mixing conditions. Decomposition of
X in point 6 of (A1) is used when we estimate the difference of [ ] and

∑ , † which appear in an asymptotic representation of .
−1
We further assume conditions for the sampling scheme. Let =

,
,
max | , − −1 | and = min | , − −1 |. For ∈ (0, 1 2), let be the
⁄
,

,
′′
′
set of all sequences {[ , )} of invervals on [0, ] satisfying
,
,
ℕ,1≤≤
′′
′
′′
′
′
′′
ℕ ⊂ ℕ, [ , 1 , , 1 ) ∩ [ , 2 , , 2 ) = ∅ for , ≠ , inf ( 1− ( − )) > 0
2
,
1
,
n,l
′
′′
and sup , ( 1− ( − )) > ∞.
,
,

[A2] There exist ∈ (0, 1 2), > 0, ̇(0, 1] and positive-valued stochastic
⁄
̇
processes { } ∈ [0, ], = 1,2 such that sup ≠ (| − |/| − | ) < ∞

⁄
−1
̇
almost surely, −1 2+ ( ) → 0 and

−1
⁄

,
′
′′
′
′′
−1 2+ max | 1− ( − ) #{; [ −1 , , ) ⊂ [ , )} − ′ | (3.2)
,

,

,
,
1≤≤ ,

Converges to zero in probability as → ∞ for {[ , )} 1≤≤ ,∈ℕ ∈
′′
′
,

,
and = 1,2. Moreover, ( 1− ) ⋁( −1− −1 ) → 0 for any > 0.

Condition [A2] is about the law of large numbers of , in each local
interval[ , ). This conditions ensures that the intensity of observation
′
′′
,
,
−1 2+ −1
⁄
count converges to some intensity with the order ( ) .

140 | I S I W S C 2 0 1 9