mixing sequences of random variables

Anais da Academia Brasileira de Ciências (2006) 78(4): 615-621 (Annals of the Brazilian Academy of Sciences) ISSN 0001-3765 www.scielo.br/aabc

Marcinkiewicz strong laws for linear statistics of ρ ∗ -mixing sequences of random variables GUANG-HUI CAI Department of Mathematics and Statistics, Zhejiang Gongshang University, Hangzhou 310035 P. R. China Manuscript received on April 4, 2006; accepted for publication on August 1st, 2006; presented by M ANFREDO DO C ARMO ABSTRACT

Strong laws are established for linear statistics that are weighted sums of a random sample. We show extensions of the Marcinkiewicz-Zygmund strong laws under certain moment conditions on both the weights and the distribution. These not only generalize the result of Bai and Cheng (2000, Statist Probab Lett 46: 105– 112) to ρ ∗ -mixing sequences of random variables, but also improve them. Key words: ρ ∗ -mixing, Marcinkiewicz-Zygmund strong laws, weighted sums.

1

INTRODUCTION

As Bai and Cheng (2000) remarked, many useful linear statistics based on a random sample are weighted sums of i.i.d. random variables. Examples include least-squares estimators, nonparametric regression function estimators and jackknife estimates, among others. In this respect, studies of strong laws for these weighted sums have demonstrated significant progress in probability theory with applications in mathematical statistics. But a random sample is often dependent. So we want to know if the results obtained for i.i.d. random variables are still true for ρ ∗ -mixing sequences of random variables. Let S, T ⊂ N be nonempty and define F S = σ (X k , k ∈ S), and the maximal correlation coefficient ρn∗ = sup corr ( f, g) where the supremum is taken over all (S, T ) with dist (S, T ) ≥ n and all f ∈ L 2 (F S ), g ∈ L 2 (FT ) and where dist (S, T ) = inf x∈S,y∈T |x − y|. A sequence of random variables {X n , n ≥ 1} on a probability space {, F, P} is called ∗ ρ -mixing if lim ρn∗ < 1. (1.1) n→∞

E-mail: [email protected] AMS Classification: 60F15; 62G05

An Acad Bras Cienc (2006) 78 (4)

616

GUANG-HUI CAI

As for ρ ∗ -mixing sequences of random variables, one can refer to Bryc and Smolenski (1993), who found bounds for the moments of partial sums for a sequence of random variables satisfying (1.1), and to Peligrad (1996) for CLT, Peligrad (1998) for invariance principles, Peligrad and Gut (1999) for the Rosenthal type maximal inequality, Utev and Peligrad (2003) for invariance principles of nonstationary sequences. The main purpose of this paper is to establish the MarcinkiewiczZygmund strong laws for linear statistics of ρ ∗ -mixing sequences of random variables. The results obtained (see Theorem 2.1 and Corollary 2.1) not only generalize the result of Bai and Cheng (2000) to ρ ∗ -mixing sequences of random variables, but also improve them. In Theorem 2.2 of Bai and Cheng (2000), they believe the choice of bn can hardly be improved in view of Cuzick (1995, Lemma 2.1), but now we improve the choice of bn using a new method.

2 THE MARCINKIEWICZ-ZYGMUND STRONG LAWS

Throughout this paper, C will represent a positive constant though its value may change from one appearance to the next, and an = O(bn ) will mean an ≤ Cbn . In order to prove our results, we need the following lemma. L EMMA 2.1. (Utev and Peligrad, 2003). Let {X i , i ≥ 1} be a ρ ∗ -mixing sequence of random variables, E X i = 0, E|X i | p < ∞ for some p ≥ 2 and for every i ≥ 1. Then there exists C = C( p), such that E max | 1≤k≤n

k

Xi | ≤ C p

i=1

n

E|X i | + p

n

i=1

p/2 E X i2

.

i=1

T HEOREM 2.1. Let {X, X i , i ≥ 1} be a ρ ∗ -mixing sequence of identically distributed random n ani X i , n ≥ 1, where the weights {ani , 1 ≤ i ≤ n, n ≥ 1} are random variables variables, Tn = i=1

which are independent of {X i , i ≥ 1} (the case of deterministic weights is included). Suppose that n for some α with 0 < α < 2 we have that i=1 |ani |α = O(n) almost surely. If 1 < α < 2, we 1 1 assume additionally that E X = 0. Set bn = n α (log n) γ . We assume that for some h, γ > 0, we have E exp(h|X |γ ) < ∞.

(2.0)

Then ∀ε > 0,

∞ n=1

n −1 P max |T j | > εbn < ∞. 1≤ j≤n

P ROOF. ∀i ≥ 1, define X i(n) = X i I (|X i | ≤ bn ), T j(n) =


(2.1)

j ani X i(n) − Eani X i(n) , then ∀ε > 0,

i=1

LINEAR STATISTICS OF ρ ∗ -MIXING SEQUENCES OF RANDOM VARIABLES

617

we have P

max |T j | > εbn

1≤ j≤n

≤P

max |X j | > bn + P

1≤ j≤n

i=1

+

j

Eani X i(n) |

> εbn

(2.2)

i=1

1≤ j≤n

bn−1 By

1≤ j≤n

|T j(n)

j max |X j | > bn + P max |T j(n) | > εbn − max | Eani X i(n) | .

First we show that

n

max

1≤ j≤n

≤P

1≤ j≤n

j (n) max Eani X i → 0, 1≤ j≤n

i=1

n → ∞.

as

(2.3)

i=1

|ani |α = O(n) and Hölder inequality, ∀1 ≤ k < α, then n

|ani | ≤ k

n

i=1

αk α−k n α |ani | 1 ≤ Cn. k αk

i=1

(2.4)

i=1

When 1 < α < 2, using E X = 0, (2.4), Markov inequality and (2.0), when n → ∞, then j

bn−1 max | 1≤ j≤n

≤ bn−1

Eani X i(n) |

i=1

n

E|ani X i |I (|X i | > bn )

i=1

= bn−1

n

|ani |E|X |I (|X | > bn )

i=1

≤ Cbn−1 n E|X |I (|X | > bn ) =

Cbn−1 n

≤ Cbn−1 n ≤ Cbn−1 n ≤ Cbn−1 n

∞ k=n ∞ k=n ∞

E|X |I (bk < |X | ≤ bk+1 )

(2.5)

bk+1 P(|X | > bk ) bk+1

k=n ∞

E exp(h|X |γ ) γ exp(hbk ) 1

(k + 1) α (log(k + 1)) γ k −hk 1

γ /α

k=n

≤ Cn

− α1

(log n)− γ nn −1 1

= Cn − α (log n)− γ → 0. 1

1


618

n i=1

then

GUANG-HUI CAI

|ani |α = O(n) implies that max1≤i≤n |ani | = O(n α ). By this and Hölder inequality, ∀k ≥ α, 1

n

n

|ani | = k

i=1

|ani |α |ani |k−α ≤ Cnn

k−α α

k

= Cn α .

(2.6)

i=1

When 0 < α ≤ 1, using (2.6), Markov inequality and (2.0), when n → ∞, then bn−1

j max Eani X i(n)

1≤ j≤n

≤ bn−1 = bn−1

i=1

n i=1 n

E|ani X i |I (|X i | ≤ bn ) |ani |E|X |I (|X | ≤ bn )

i=1

≤ Cbn−1 n α E|X |I (|X | ≤ bn ) 1

= Cbn−1 n α 1

≤ Cbn−1 n α 1

n k=2 n

E|X |I (bk−1 < |X | ≤ bk )

(2.7)

bk P(|X | > bk−1 )

k=2

≤ Cbn−1 n α 1

n

bk

k=2

≤ Cbn−1 n α 1

n

E exp(h|X |γ ) γ exp(hbk−1 ) γ /α

1

k α (log k) γ (k − 1)−h(k−1) 1

k=2

≤ Cn

− α1

(log n)− γ n α 1

1

= C(log n)− γ → 0. 1

From (2.5) and (2.7), Hence (2.3) is true. From (2.2) and (2.3), it follows that for n large enough n ε P max |T j | > εbn ≤ P |X j | > bn + P max |T j(n) | > bn . 1≤ j≤n 1≤ j≤n 2 j=1

Hence we need only to prove that I =:

∞

n −1

n=1

I I =:

∞ n=1


n P |X j | > bn < ∞, j=1

n −1 P max

1≤ j≤n

|T j(n) |

ε > bn < ∞. 2

(2.8)


619

From the fact that E exp(h|X |γ ) < ∞, it follows easily that

I = =

∞

n −1 n P(|X | > bn )

n=1 ∞

P(|X | > bn )

n=1

(2.9)

∞ E exp(h|X |γ ) ≤ γ exp(hbn ) n=1

≤C

∞ n=1

1 n hn γ /α

< ∞.

By Lemma 2.1, it follows that for q ≥ 2

II ≤ C ≤C

∞ n=2 ∞

n −1 bn−q E max |T j(n) |q 1≤ j≤n

n −1 bn−q

n

n=2

q E|an j X (n) j |

+

n

j=1

2 E|an j X (n) j |

q/2

(2.10)

j=1

=: I I1 + I I2 . Let max(2, α, γ + 1) ≤ q, using (2.6), we have

I I1 = C

∞

n −1 bn−q

n=2

≤C =C ≤C

∞ n=2 ∞

n

|ani |q E|X |q I (|X | ≤ bn )

i=1 q

n −1 bn−q n α E|X |q I (|X | ≤ bn ) n

q

n −1 bn−q n α

n=2 ∞ ∞

E|X |q I (bk−1 < |X | ≤ bk )

k=2

n

−1+ αq

n

−q/α

(2.11) q (log n)−q/γ bk P(|X |

> bk−1 )

k=2 n=k

≤C

∞

q

bk

k=2

≤C

∞

q

E exp(h|X |γ ) γ exp(hbk−1 ) q

γ /α

k α (log k) γ (k − 1)−h(k−1)

< ∞.

k=2


620

GUANG-HUI CAI

By 0 < α < 2, (2.6) and q ≥ (γ + 1), we have I I2 = C ≤C

∞ n=2 ∞

n −1 bn−q

n

q/2 |ani |

2

=C ≤C ≤C

n=2 ∞ n=2 ∞

q/2

E|X |2 I (|X | ≤ bn )

i=1

2 q/2 q/2 E|X |2 I (|X | ≤ bn ) n −1 bn−q n α

n=2 ∞

−1

−q/γ

n (log n) −1

−q/γ

n (log n) −1

−q/γ

n (log n)

n=2

n k=2 n k=2 n

q/2 E|X | I (bk−1 < |X | ≤ bk ) 2

q/2 bk2 P(|X | bk2

k=2

> bk−1 )

E exp(h|X |γ ) γ exp(hbk−1 )

q/2 (2.12)

n

2 q/2 2 k α (log k) γ ≤C n (log n) exp(h(k − 1)γ /α log(k − 1)) n=2 k=2 q/2 ∞ n 2 2 −1 −q/γ −h(k−1)γ /α γ α =C n (log n) k (log k) (k − 1)

∞

−1

−q/γ

n=2

≤C ≤C

∞ n=2 ∞

k=2

−1

−q/γ

n (log n)

n

k

−2

q/2

k=2

n −1 (log n)−q/γ < ∞.

n=2

Putting (2.11) and (2.12) into (2.10) yields I I < ∞. Now we complete the prove of Theorem 2.1. |Tn | n→∞ bn

C OROLLARY 2.1. Under the conditions of Theorem 2.1, then lim

=0

P ROOF. By (2.1), we have ∞ >

∞

n −1 P max |T j | > εbn 1≤ j≤n

n=1 ∞ 2 −1 i+1

= ≥

1 1 n −1 P max |T j | > εn α (log n) γ 1≤ j≤n

i=0 n=2i ∞

1 2

1 i+1 P max |T j | > ε2 α (log 2i+1 ) γ .

i=1

1≤ j≤2i

By Borel-Cantelli Lemma, we have P


max |T j | > ε2

1≤ j≤2i

i+1 α

(log 2

i+1

1 γ

) , i.o. = 0.

a.s.


Hence

max |T j |

1≤ j≤2i

lim

i→∞

and using max

2i−1 ≤n