LIMITING BEHAVIOR OF TIIE PERTURBED EMPIRICAL DISTRIBUTION FUNCTIONS EVALUATED AT U-STATISTICS FOR STRONGLY MIXING SEQUENCES OF RANDOM VARIABLES

We prove the almost sure representation, a law of the iterated logarithm and an invariance principle for the statistic Fˆn(Un) for a class of strongly mixing sequences of random variables {Xi,i≥1}. Stationarity is not assumed. Here Fˆn is the perturbed empirical distribution function and Un is a U-statistic based on X1,…,Xn.


Introduction
Let {Xi, i 1} be a sequence of nonstationary random variables with c.d.f.'s {Fi, _ 1) defined on the real line and assume Fi-F as i--,oc for some fixed distribution function F. Also, let Fn(x be the corresponding empirical distribution function based on Xi,...,Xn, that is, n(x)-n-l = l U(X-Xi), where (.)_ , >0 0, elsewhere.
Consider the sequence of perturbed empirical distribution functions given by Fn(x) n-l -Kn(x Xi), n >_ l, x e N, i=1 Printed in the U.S.A. ()1997 by North Atlantic Science Publishing Company however, there has been much interest in the cases of dependence in probability and statistics in general and mixing conditions in particular.The latter represent degrees of weak dependence in the sense of asymptotic independence of past and distant future ([6] an..d [5]).In this connection, recently [17] has proved the asymptotic normality of Fn(Un) for the case where Xi's form an absolutely regular stationary sequence.
In this paper we study the asymptotic behavior of Fn(Un) in the case where the sequence {Xi, >_ 1} is strong mixing, which is more general then the absolutely regular case and is about the weakest mixing condition (see, e.g., [3] and [5]).More- over, we do not assume stationarity.Specifically, we give the almost sure representation, a law of the iterated logarithm and an invariance principle for Fn(Un).Thus the results obtained here extend or generalize those of [14], [10] and [17].
(2.3)Then, it follows from the definition of strong -ij and /-+. (2.4) Let A c be the complement of A. Then we have M O6[a(d)] 5 M[a(d)] , (i 0, 1).
We shall take a(n) pn, 0 < p < 1, in the remainder of this paper.
Next, we provide the following lemma as a generalization of a result of [1].
We introduce the following two sets of conditions: (A) An(tj, n)" (2.25) {F, k 1} is uniformly bounded in the neighborhood of .foo (i) {Fi, >_ 1} is twice differentiable with uniformly bounded tF:'i,, >_ 1} in the neighborhood of .
(ii) f_oox2k(x)dx < oo and there exists a 7 > 0 such that k(x)-k(-x), The almost sure representation theorem given in section 4 highly relies on the following two lemmas.
Then there exists an e > 0 such that by O(na(log log n) 1).
A similar method can be used to estimate the sums in the other cases and thus obtain (2.40).The proof of (2.41), which is analogous, is omitted.
Lemma 2.7: Under the conditions of Lemma 2.6, we have Un--mU)+R n (2.51) where R n -O(n-l(loglogn) Proof: The proof of the lemma follows by applying Lemma 2.6 and the approach of Theorem 1 of [21].Lemma 2.8: Let {Yni, 1 <_ <_ n,n >_ 1} be a sequence of strong mixing random variables with mean O. Suppose for any n, rn such that rn >_ rn and any J C {1,..., with Card J rn j.J for some r 2 > 0.Then, the process {Yni, l <-<n,n >-l} obeys the law of the iterated logarithm if the following conditions are satisfied for some 6 and ' such that sup max E IYnil 2 +5-M<c  Proof: The lemma was proved by [7], Lemma 5.6 for a sequence of random variables satisfying the strong mixing condition with mean zero.Lemma 2.9: Let [Yni, 1 <_ <_ n,n >_ 1} be a sequence of random variables satisfying the strong mixing condition with coefficient a(n).
Almost Sure Representation and a Law of Iterated Logarithm for F.(U.) Theorem 4.1: Suppose that Vi >_ 1, F'() exists and is finite and let {Fi, >_ 1) and k satisfy (A).Furthermore, suppose the assumptions of Lemma where Xni is defined as in (3.5)Moreover, if {Fi} and k satisfy (B), then we may replace as n---oo and 0 < r < g O(an) b$10(a2n) in (4.2).
Proof: Using Theorem 4.2, the proof follows easily by applying Lemma 2.8.
Next, let W be a random Nnction of [0, 1] defined as
i=1Limiting Behavior of the Perturbed Empirical Distribution Functions 15 2.6are satisfied, a(n) pn, for O < p < 1, and a n o(n-lloglogn).Then