An example of nonquenched convergence in the conditional central limit theorem for partial sums of a linear process


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1 A example of oqueched covergece i the coditioal cetral limit theorem for partial sums of a liear process Dalibor Volý ad Michael Woodroofe Abstract A causal liear processes X,X 0,X is costructed for which the coditioal distributios of stadardized partial sums S = X + + X give X,X 0 coverge i probability to the stadard ormal distributio, but do ot coverge w.p.. Itroductio Let X,X 0,X, deote a strictly statioary sequece defied o a probability space (Ω,A,) ad adapted to a filtratio F k. Suppose that the X k have mea 0 ad fiite variace; ad let S = X + +X ad σ 2 = E(S 2 ). With this otatio the Coditioal Cetral Limit Questio may be stated: Whe do the coditioal distributios of a S /σ coverge i probability to the stadard ormal distributio; that is, whe does the Levy distace betwee the coditioal distributio ad the stadard ormal coverge i probability zero? Two sets of ecessary ad sufficiet coditios may be foud i 3 ad 0. Oe ca also ask: Whe is the covergece queched; that is, whe do the coditioal distributios coverge almost surely? I a Markov Chai settig, this meas that the covergece takes place for almost every (with respect to the statioary measure) startig poit. It has bee show that several importat classical limit theorems are queched. See ad (for more recet research, e.g.) 5, 2, ad. Dalibor Volý Michael Woodroofe
2 2 D. Volý ad M. Woodroofe For a causal liear process X k = i=0 a i ξ k i, () where a 0,a, are square summable ad ξ,ξ 0,ξ, are i.i.d. with mea 0 ad variace oe, let F = σ{,ξ,ξ }. The S = i=0 b i+ b i ξ i + i= b i ξ i, where b = a a. It follows that E(S F 0 ) = i=0 b i+ b i ξ i ad S E(S F 0 ) = i= b iξ i are idepedet. The, lettig deote the orm i L 2 () ad τ 2 = b b2, ad E(S F 0 ) 2 = i=0 b i+ b i 2 = ν 2 say, σ 2 = EE(S F 0 ) 2 +E{S E(S F 0 ) 2 } = ν 2 + τ2. With this otatio, Wu ad Woodroofe 0 showed that if lim σ 2 =, the the coditioal distributio fuctio of S /σ give F 0 coverges i probability to the stadard ormal distributio iff ν 2 = o(σ2 ). Here a causal liear process with absolutely summable coefficiets is costructed for which the coditioal distributios of S /σ coverge to Φ i probability but ot with probability oe. The summability of coefficiets meas that Haa s coditio 7, i 0 E(X 0 F i ) E(X 0 F i ) <, is satisfied ad, therefore, that the (weak) ivariace priciple holds (see 4). 2 The prelimiaries The first step is to develop a ecessary ad sufficiet coditio for queched covergece. The proof of the lemma below uses the Covergece of Types Theorem (8, p. 203): Let Y ad Z be radom variables of the form Y = α Z + β, where α, β R are costats. If Y Y ad Z Z, where Z odegeerate, the α ad β coverge to limits α ad β ad Y = dist αz + β. Lemma For a causal liear process () for which σ ad ν = o(σ ), the coditioal distributio of S /σ give F 0 coverges to the stadard ormal distributio w.p. iff lim E(S F 0 ) = 0 w.p.. (2) σ roof. Suppose first that ν = o(σ ) i which case τ 2 /σ2, ad let F deote the (ucoditioal) distributio fuctio of S E(S F 0 )/τ. The S σ z E(S z F F 0 ) 0 = F, σ τ
3 Noqueched covergece i the coditioal cetral limit theorem 3 by idepedece. It is show below that F coverges weakly to the stadard ormal distributio Φ. The sufficiecy of (2) for almost sure covergece of the coditioal distributio fuctio is the obvious. Coversely, if the coditioal distributios coverge almost surely to the stadard ormal distributio, the E(S F 0 )/σ 0 w.p., by the Covergece of Types Theorem, applied coditioally. That F Φ follows from Theorem 2. ad Corollary 2. of 9. The argumet is sketched here for completeess. By the Lideberg Feller Coditio, it is sufficiet to show that lim b 2 i 0 i τ 2 = 0. (3) Suppose that the imum is attaied at i ad (temporarily) that i 2; ad let A 2 = i=0 a2 i. If k 2, the b 2 i b 2 i +k = k k i= (b i + j b i + j)(b i + j + b i + j) a 2 i + j k (b i + j + b i + j) 2 2Aτ. So, for ay m 2, mb2 i m k= b2 i +k + 2Amτ τ 2 + 2Amτ ad, therefore b 2 i τ 2 m + 2A τ. The same iequality may be obtaied if i 2 by a dual argumet i which k is replaced by k, ad (3) follows by lettig ad m i that order. Lemma 2 For a causal liear process (): If a 0,a,a 2, are absolutely summable ad b := i=0 a i 0, the σ 2 b 2 ad ν 2 = o(σ 2 ). roof. The proof uses a differet expressio for E(S F 0 ), E(S F 0 ) = a j ζ j + a j ζ j ζ j, j=+ where ζ j = ξ j+ + + ξ 0. Thus, τ 2 = b b2 b2 ad E(S F 0 ) a j j + a j = o( ) = o(σ ). j=+ The lemma follows directly.
4 4 D. Volý ad M. Woodroofe 3 The example The mai result follows. Theorem There are oegative summable coefficiets a 0,a,a 2, for which ν 2 = o(σ 2 ) but (2) fails. roof. By Lemma 2, it suffices to costruct positive summable coefficiets a 0,a,a 2, for which (2) fails. We cosider coefficiets of the form a j = 2 j j, (4) for j, ad a i = 0 if i / {, 2, }, where 0 = 0 < < 2 < is a sequece of positive itegers costructed below. It is clear that a sequece of the form (4) is summable. For sequeces of the form (4), E(S F 0 ) = j a j ζ j + j > a j ζ j ζ j, where ζ = ξ ξ 0, as above. So, for k < k where E(S F 0 ) = I k ()+II k (), k I k () = II k () = a j ζ j + a k ζ k ζ k, (5) a j ζ j ζ j, j=k+ ad the empty sum is to be iterpreted as zero whe k =. The specificatio of the itegers i depeds o the followig claim: There are itegers 0 < < 2 < for which I k () k < k > 2 k+ > ( ) k+ (6) 2 for all k. The proof of the claim, i tur, depeds o the followig observatio: Let J(N,) = ζ N ζ N = ξ N+ + + ξ N+ for N, so that I k () = k a j ζ j +a k J( k,) for k. The the joit distributio of J(N,), N, is the same as the joit distributio of ζ = ξ + + ξ, N. It the follows from the Law of the Iterated Logarithm that 0 <N J(N, ) > 8 = 0 <N ζ > 8
5 Noqueched covergece i the coditioal cetral limit theorem 5 as N. So the right side is at least 3/4 all sufficietly large N, ad the existece of follows (sice I ()=J(,)/2). Now, let k 2 ad suppose that,, k have bee costructed. The λ k ca be chose so that k ( ) k+2 a j ζ j > λ k. 2 As above, k <N = J(N, ) > 2 k k (λ k + 2 k+ ) k <N ζ > 2 k k (λ k + 2 k+ ), which approaches as N by the Law of the Iterated Logarithm. The existece of k i (6) follows directly from the last two displays ad (5). The existece of the sequece the follows from mathematical iductio (the existece of,, k implies that of k ). The ext step is to boud the term II k (). By Doob s (953) imal iequality, E k ζ k 2 for all. The So So, E II k () k < k a j E j=k+ j=k+ II k () > 2 k k < k ζ j ζ j k < k 2 k 2 j j 2 k. 2 2k. That (2) fails for this costructio may be see as follows: Clearly, E(S F 0 ) k < k k < k I k () II k (). k < k k < k k E(S F 0 ) 2 k I k () 2 k+ k < k + II k () 2 k k < k for k 2. It follows that E(S F 0 ) 2, k ifiitely ofte = 0 k < k 2 2 k ad, therefore, that limsup E(S F 0 )/ = w.p..
6 6 D. Volý ad M. Woodroofe Refereces A. N. Borodi ad I. A. Ibragimov, Limit theorems for fuctioals of radom walks, Trudy Mat. Ist. Steklov. 95 (994). Trasl. ito Eglish: roc. Steklov Ist. Math. 95 (995), o.2. 2 C. Cuy, oitwise ergodic theorems with rate ad applicatio to limit theorems for statioary processes, arxiv: v, J. Dedecker ad F. Merlevède, Necessary ad sufficiet coditios for the coditioal cetral limit theorem, A. rob. 32 (2002), J. Dedecker, F. Merlevède, ad D. Volý, O the weak ivariace priciple for o adapted sequeces uder projective criteria, Joural of Theoretical robability 20 (2007), Y. Derrieic ad M. Li, The cetral limit theorem for Markov chais with ormal trasitio operators started at a poit, robab. Theory Relat. Fields 9 (200), J. Doob, Stochastic rocesses, Wiley, E. J. Haa, Cetral limit theorems for time series regressio, Z. Wahrscheilichkeitstheorie verw. Geb. 26 (973), M. Loeve, robability Theory, Va Nostrad, M. eligrad ad S. Utev, Cetral limit theorem for liear processes, A. rob. 25 (997), WeiBiao Wu ad M. Woodroofe, Martigale approximatios for sums of statioary processes, A. rob. 32 (2004), Ou Zhao ad M. Woodroofe, Law of the iterated logarithm for statioary processes, A. rob. 37, (2007),
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