Marcinkiewicz strong laws for linear statistics of ρ ∗-mixing sequences of random variables

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.


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 signifi cant 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 defi ne F S = σ (X k , k ∈ S), and the maximal correlation coeffi cient ρ * n = 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 (F T ) and where dist (S, T ) = inf x∈S,y∈T |x − y|.
A sequence of random variables 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 Marcinkiewicz-Zygmund 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 b n can hardly be improved in view of Cuzick (1995, Lemma 2.1), but now we improve the choice of b n using a new method.

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 In order to prove our results, we need the following lemma.
COROLLARY 2.1.Under the conditions of Theorem 2.1, then lim n→∞ By Borel-Cantelli Lemma, we have REMARK 2.1.Corollary 2.1 generalizes the Theorem 2.2 of Bai and Cheng (2000) to ρ * -mixing sequences of random variables and the restricton of b n in Corollary 2.1 is weaker than the restricton of b n in Theorem 2.2 of Bai and Cheng (2000).