An example of non-quenched convergence in the conditional central limit theorem for partial sums of a linear process Dalibor Voln´y and Michael Woodroofe
Abstract A causal linear processes · · · X−1 , X0 , X1 · · · is constructed for which the conditional distributions of standardized partial sums Sn = X1 + · · · + Xn given · · · X−1 , X0 converge in probability to the standard normal distribution, but do not converge w.p. 1.
1 Introduction Let · · · X−1 , X0 , X1 , · · · denote a strictly stationary sequence defined on a probability space (Ω, A , P) and adapted to a filtration Fk . Suppose that the Xk have mean 0 and finite variance; and let Sn = X1 + · · · + Xn and σ2n = E(Sn2 ). With this notation the Conditional Central Limit Question may be stated: When do the conditional distributions of a Sn /σn converge in probability to the standard normal distribution; that is, when does the Levy distance between the conditional distribution and the standard normal converge in probability zero? Two sets of necessary and sufficient conditions may be found in [3] and [10]. One can also ask: When is the convergence quenched; that is, when do the conditional distributions converge almost surely? In a Markov Chain setting, this means that the convergence takes place for almost every (with respect to the stationary measure) starting point. It has been shown that several important classical limit theorems are quenched. See [1] and (for more recent research, e.g.) [5], [2], and [11]. Dalibor Voln´y e-mail:
[email protected] Michael Woodroofe
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D. Voln´y and M. Woodroofe
For a causal linear process ∞
Xk = ∑ ai ξk−i ,
(1)
i=0
where a0 , a1 , · · · are square summable and · · · ξ−1 , ξ0 , ξ1 , · · · are i.i.d. with mean 0 and variance one, let Fn = σ{· · · , ξn−1 , ξn }. Then ∞
n
i=0
i=1
Sn = ∑ [bi+n − bi ]ξ−i + ∑ bn−i ξi , where bn = a0 + · · · + an . It follows that E(Sn |F0 ) = ∑∞ i=0 [bi+n − bi ]ξ−i and Sn − E(Sn |F0 ) = ∑ni=1 bn−i ξi are independent. Then, letting k ·k denote the norm in L2 (P) and τ2n = b20 + · · · + b2n−1, ∞
kE(Sn |F0 )k2 = ∑ [bi+n − bi ]2 = ν2n
say,
i=0
and
σ2n = E[E(Sn |F0 )2 ] + E{[Sn − E(Sn|F0 )]2 } = ν2n + τ2n .
With this notation, Wu and Woodroofe [10] showed that if limn→∞ σ2n = ∞, then the conditional distribution function of Sn /σn given F0 converges in probability to the standard normal distribution iff ν2n = o(σ2n ). Here a causal linear process with absolutely summable coefficients is constructed for which the conditional distributions of Sn /σn converge to Φ in probability but not with probability one. The summability of coefficients means that Hannan’s condition [7], ∑i≥0 kE(X0 |Fi ) − E(X0 |Fi−1 )k < ∞, is satisfied and, therefore, that the (weak) invariance principle holds (see [4]).
2 The preliminaries The first step is to develop a necessary and sufficient condition for quenched convergence. The proof of the lemma below uses the Convergence of Types Theorem ([8], p. 203): Let Yn and Zn be random variables of the form Yn = αn Zn + βn , where αn , βn ∈ R are constants. If Yn ⇒ Y and Zn ⇒ Z, where Z non-degenerate, then αn and βn converge to limits α and β and Y =dist αZ + β. Lemma 1 For a causal linear process (1) for which σn → ∞ and νn = o(σn ), the conditional distribution of Sn /σn given F0 converges to the standard normal distribution w.p. 1 iff 1 lim E(Sn |F0 ) = 0 w.p. 1. (2) n→∞ σn Proof. Suppose first that νn = o(σn ) in which case τ2n /σ2n → 1, and let Fn denote the (unconditional) distribution function of [Sn − E(Sn|F0 )]/τn . Then Sn σn z − E(Sn|F0 ) P ≤ z|F0 = Fn , σn τn
Non-quenched convergence in the conditional central limit theorem
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by independence. It is shown below that Fn converges weakly to the standard normal distribution Φ. The sufficiency of (2) for almost sure convergence of the conditional distribution function is then obvious. Conversely, if the conditional distributions converge almost surely to the standard normal distribution, then E(Sn |F0 )/σn → 0 w.p. 1, by the Convergence of Types Theorem, applied conditionally. That Fn ⇒ Φ follows from Theorem 2.1 and Corollary 2.1 of [9]. The argument is sketched here for completeness. By the Lindeberg–Feller Condition, it is sufficient to show that b2 lim max 2i = 0. (3) n→∞ 0≤i≤n τn Suppose that the maximum is attained at in and (temporarily) that in ≤ 21 n; and let 1 2 A2 = ∑∞ i=0 ai . If k ≤ 2 n, then b2in − b2in +k =
k
∑ (bin + j−1 − bin+ j )(bin + j−1 + bin+ j )
j=1
v u u ≤t
v u u t
k
k
∑ a2in+ j ∑ (bin + j−1 + bin+ j )2 ≤ 2Aτn .
i=1
j=1
2 2 So, for any m ≤ 12 n, mb2in ≤ ∑m k=1 bin +k + 2Amτn ≤ τn + 2Amτn and, therefore
b2in 1 2A ≤ + . 2 τn m τn The same inequality may be obtained if in ≥ 21 n by a dual argument in which k is replaced by −k, and (3) follows by letting n → ∞ and m → ∞ in that order. Lemma 2 For a causal linear process (1): If a0 , a1 , a2 , · · · are absolutely summable 2 2 2 2 and b := ∑∞ i=0 ai 6= 0, then σn ∼ b n and νn = o(σn ). Proof. The proof uses a different expression for E(Sn |F0 ), ∞
n
E(Sn |F0 ) =
∑ a jζ j +
j=1
∑
j=n+1
a j [ζ j − ζ j−n],
where ζ j = ξ− j+1 + · · · + ξ0 . Thus, τ2n = b20 + · · · + b2n−1 ∼ b2 n and n
kE(Sn |F0 )k ≤
∑ aj
j=1
The lemma follows directly.
p
∞
j+
∑
√ √ a j n = o( n) = o(σn ).
j=n+1
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D. Voln´y and M. Woodroofe
3 The example The main result follows. Theorem 1 There are non-negative summable coefficients a0 , a1 , a2 , · · · for which ν2n = o(σ2n ) but (2) fails. Proof. By Lemma 2, it suffices to construct positive summable coefficients a0 , a1 , a2 , · · · for which (2) fails. We consider coefficients of the form an j =
1
√ 2j n
,
(4)
j−1
for j ≥ 1, and ai = 0 if i ∈ / {n1 , n2 , · · · }, where 0 = n0 < n1 < n2 < · · · is a sequence of positive integers constructed below. It is clear that a sequence of the form (4) is summable. For sequences of the form (4), E(Sn |F0 ) = ∑n j ≤n an j ζn j + ∑n j >n an j [ζn j − ζn j −n ], where ζn = ξ−n+1 + · · · + ξ0 , as above. So, for nk−1 ≤ n < nk E(Sn |F0 ) = Ik (n) + IIk (n), where k−1
Ik (n) =
∑ an j ζn j + ank [ζnk − ζnk −n ],
j=1 ∞
IIk (n) =
∑
j=k+1
(5)
an j [ζn j − ζn j −n ],
and the empty sum is to be interpreted as zero when k = 1. The specification of the integers ni depends on the following claim: There are integers 0 < n1 < n2 < · · · for which k+1 Ik (n) 1 k+1 √ >2 P max > 1− nk−1 ≤n 8 = P max √ > 8 → 1 P max √ 0≤n
