Download An Example of Non-Quenched Convergence In the Conditional

An Example of Non-Quenched Convergence
In the Conditional Central Limit Theorem
For Partial Sums of a Linear Process
Dalibor Volný 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.
Key words and phrases: Convergence of types; law of the iterated logarithm; maximal inequality.
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 σn2 = 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 Lévy distance between the conditional distribution and the standard normal converge in probability
to zero? Two sets of necessary and sufficient conditions may be found in [3] and [9].
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 [10]. Here it is shown that for a
stationary linear process with independent innovations and summable coefficients,the CLT
need not be quenched. The summability of coefficients means that the Hannan’s condi-
1
tion
P
i∈Z
kE(X0 |Fi ) − E(X0 |Fi−1 )k < ∞ is satisfied, hence, if X0 is F∞ -measurable and
E(X0 |F−∞ ) = 0, the CLT and the (weak) invariance principle take place (see [4]).
For a causal linear process
Xk =
∞
X
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 Sn = X1 + · · · + Xn , σn2 = E(Sn2 ), and Fn = σ{· · · , ξn−1 , ξn }. Then
Sn =
∞
X
[bi+n − bi ]ξ−i +
i=0
n
X
bn−i ξi ,
i=1
P∞
where bn = a0 + · · · + an . It follows that E(Sn |F0 ) = i=0 [bi+n − bi ]ξ−i and Sn − E(Sn |F0 ) =
Pn
2
2
2
i=1 bn−i ξi are independent. Then, letting k · k denote the norm in L (P ) and τn = b0 +
· · · + b2n−1 ,
2
kE(Sn |F0 )k =
∞
X
[bi+n − bi ]2 = νn2
say,
i=0
and
σn2 = E[E(Sn |F0 )2 ] + E{[Sn − E(Sn |F0 )]2 } = νn2 + τn2 .
With this notation, Wu and Woodroofe [9] showed that if limn→∞ σn2 = ∞, then the conditional distribution function of Sn /σn given F0 converges in probability to the standard
normal distribution iff νn2 = o(σn2 ). Here it is shown that “convergence in probability” cannot be replaced by “convergence w.p.1” (i.e. the limit theorem is not “quenched”) without
imposing further conditions.
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], pp. 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
E(Sn |F0 ) = 0 w.p.1.
n→∞ σn
lim
2
(2)
Proof. Observe that τn2 /σn2 → 1 as n → ∞, since νn2 = o(σn2 ), 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
by independence. It is shown below that the 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.
It remains to show that Fn ⇒ Φ, and for this it is sufficient to show that
b2i
= 0.
n→∞ 0≤i≤n τn2
lim max
(3)
See, for example, [7], p 153. Suppose that the maximum is attained at in and (temporarily)
P
1
2
that in ≤ 12 n; and let A2 = ∞
i=0 ai . If k ≤ 2 n, then
b2in − b2in +k =
k
X
(bin +j−1 − bin +j )(bin +j−1 + bin +j )
j=1
v
u k
uX
a2
≤t
in +j
i=1
So, for any m ≤ 12 n, mb2in ≤
v
u k
uX
t (bi
.
n +j−1
+ bin +j )2 ≤ 2Aτn
j=1
Pm
2
k=1 bin +k
+ 2Amτn ≤ τn2 + 2Amτn and, therefore
b2in
1
2A
≤
+
.
2
τn
m
τn
The same inequality may be obtained if in ≥ 12 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 and
P
2
2
2
2
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 ),
E(Sn |F0 ) =
n
X
aj ζj +
j=1
∞
X
aj [ζj − ζj−n ],
j=n+1
where ζj = ξ−j+1 + · · · + ξ0 . Thus, τn2 = b20 + · · · + b2n−1 ∼ b2 n and
kE(Sn |F0 )k ≤
n
X
j=1
aj
∞
X
p
√
√
j+
aj n = o( n) = o(σn ).
j=n+1
♦
The lemma follows directly.
3
3
The Example
The main result follow.
Theorem 1 There are non-negative summable coefficients a0 , a1 , a2 , · · · for which νn2 =
o(σn2 ) but (2) fails.
Proof. By Lemma 2, it suffices to construct positive summable coefficients a0 , a1 , a2 , · · ·
(2) fails. We consider coefficients of the form
anj =
1
√
2j n
,
(4)
j−1
for j ≥ 1, and ai = 0 if i ∈
/ {n1 , n2 , · · · }, where 0 < n1 < n2 < · · · is a sequence of positive
integers constructed below. It is clear that a sequence of the form (4) is summable.
P
P
For sequences of the form (4), E(Sn |F0 ) = nj ≤n anj ζnj + nj >n anj [ζnj − ζnj −n ]. So, for
nk−1 ≤ n < nk
E(Sn |F0 ) = Ik (n) + IIk (n),
where
Ik (n) =
k−1
X
anj ζnj + ank [ζnk − ζnk −n ],
j=1
IIk (n) =
∞
X
anj [ζnj − ζnj −n ],
j=k+1
and an empty sum is to be interpreted as zero.
The specification of the integers ni depends on the following claim: There are integers
0 < n1 < n2 < · · · for which
P
max
nk−1 ≤n<nk
k+1
Ik (n)
1
k+1
√ >2
>1−
2
n
(5)
for all k ≥ 1. To see this let n0 = 1 and suppose that n0 , · · · , nk−1 have been constructed.
Then there is a λk for which
"
P |
k−1
X
#
anj ζnj | > λk
j=1
k+2
1
≤
.
2
Let J(N, n) = [ζN − ζN −n ] for 1 ≤ n ≤ N , so that Ik (n) =
Pk−1
j=1
anj ζnj + ank J(nk , n)
Then the joint distribution of J(N, n), 1 ≤ n ≤ N is the same as the joint distribution of
ζn0 = ξ1 + · · · + ξn , 1 ≤ n ≤ N . So, for any λ > 0,
J(N, n) p k
k+1
√
> 2 nk−1 (λk + 2 )
P
max
nk−1 ≤n<N
n
,
p
ζn0
k+1
=P
max √ > 2k nk−1 (λk + 2 )
nk−1 ≤n<N
n
4
which approaches 1 as N → ∞ by the Law of the Iterated Logarithm. The existence of nk
in (5) follows directly, and the existence of the sequence by mathematical induction.
The next step is to bound the term IIk (n). By Doob’s (1953) maximal inequality,
√
E[maxk≤n |ζk |] ≤ 2 n for all n. Then
E
max
nk−1 ≤n<nk
∞
X
|IIk (n)| ≤
anj E
max
nk−1 ≤n<nk
j=k+1
∞
X
|ζnj
− ζnj −n ||
√
2 nk
1
≤ k−1 .
≤
√
j
2 nj−1
2
j=k+1
So
P
max
nk−1 ≤n<nk
k
|IIk (n)| > 2
≤
1
22k−1
.
That (2) fails for this construction may be seen as follows: Clearly,
max
nk−1 ≤n<nk
Ik (n)
|IIk (n)|
1
√ E(Sn |F0 ) ≥ max
√ − max
.
√
nk−1 ≤n<nk
nk−1 ≤n<nk
nk−1
n
n
So,
P
max
nk−1 ≤n<nk
1
Ik (n)
k
k+1
√ E(Sn |F0 ) ≤ 2 ≤ P
√ ≤2
max
nk−1 ≤n<nk
n
n
2
k
+P
max |IIk (n)| ≥ 2 ≤ k
nk−1 ≤n<nk
2
for k ≥ 2. It follows that
P
max
1
k
√ E(Sn |F0 ) ≤ 2 , infinitely often] = 0
nk−1 ≤n<nk
n
√
and, therefore, that lim supn→∞ E(Sn |F0 )/ n = ∞ w.p.1.
♦
References
[1] Borodin, Andrei Nokolaevich and Ibragimov, Ildar Abdullovich (1994). Limit theorems
for functionals of random walks. Trudy Mat. Inst. Steklov., 195. Transl. into English:
Proc. Steklov Inst. Math. (1995), 195, no.2.
[2] Cuny, Christophe (2009). Pointwise ergodic theorems with rate and application to limit
theorems for stationary processes. arXiv:0904.0185v1.
[3] Dedecker, Jérôme and Merlevède, Florence (2002). Necessary and sufficient conditions
for the conditional central limit theorem Ann. Prob., 32, 1044-1081.
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[4] Dedecker, Jérôme, Merlevède, Florence and Volný, Dalibor (2007). On the weak invariance principle for non adapted sequences under projective criteria. Journal of Theoretical
Probability, 20, 971-1004.
[5] Derriennic, Yves and Lin, Michael (2001). The central limit theorem for Markov chains
with normal transition operators started at a point. Probab. Theory Relat. Fields, 119,
508-528.
[6] Doob, Joe (1953). Stochastic Processes. Wiley
[7] Hájek, J, and Z. Šidák (1967). The Theory of Rank Tests. Academic Press.
[8] Loève, Michael (1963). Probability Theory. Van Nostrand.
[9] Wu, Wei-Biao and Michael Woodroofe (2004). Martingale approximations for sums of
stationary processes. Ann. Prob., 32, 1674-1690.
[10] Zhao, Ou and Michael Woodroofe (2007). Law of the iterated logarithm for stationary
processes. Ann. Prob., 37, 127-142.
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