Page 29 - Contributed Paper Session (CPS) - Volume 4

CPS2107 Junfan Tao et al.
the sequential estimates and the stopping times. Here, ‖. ‖ denotes the L 2
norm and ⌊⌋ the integer part of a for a > 0. Let Z− = {...,−2,−1,0}.

Theorem 2. Let , and P be a probability measure on
(Ω,F). For ω = (ωn) ∈ Ω, define the coordinate process xn (ω) = ωn and assume that
2
xn is stationary and ergodic. Let Fn = σ [xk : k ≤ n] and ξn = h(··· ,xn−1,xn) be L -
Z −
random variables with a common h : R → R. If

, (14)
then
E [ξn] = 0,
and the series

(15)
converges absolutely. When ν > 0, and , then

⌊⌋ /√ ⇒ ( → ∞)

in the sense of D[0,∞), where W is a Brownian Motion.
Remark 3. Note that Theorem 2 is a simple extension of Theorem 19.1 in
Billingsley(1999, p.197). Unlike Billingsley’s theorem, a process to which we
apply the functional central limit theorem can be di erent from a process
generating ltration in our theorem.

3. Result
The previous section’s results lead to the following lemma.
Lemma 4. Suppose x0 has the stationary distribution in (1). Let

(16)
and

. (17)
As c ↑ ∞, we have

(18)
in the sense of D[0,∞), where W ,W and W are standard Brownian
(2)
(1)
(3)
motions with correlation matrix

. (19)

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