The Law of the Iterated Logarithm and Marcinkiewicz Law of Large Numbers for B-Valued U-Statistics 1
نویسنده
چکیده
Suppose that B is a separable Banach space and (S, 6 a, P) a probability space. H is a measurable symmetric kernel function from S" into B. In this paper we shall further study some limit theorems for B-valued U-statistics U",,H based on P and H. Special attention is paid upon the Marcinkiewicz type law of large numbers and the law of the iterated logarithm. Our results can be regarded as extensions of corresponding results for sums of independent B-valued random variables to U-statistics.
منابع مشابه
MARCINKIEWICZ-TYPE STRONG LAW OF LARGE NUMBERS FOR DOUBLE ARRAYS OF NEGATIVELY DEPENDENT RANDOM VARIABLES
In the following work we present a proof for the strong law of large numbers for pairwise negatively dependent random variables which relaxes the usual assumption of pairwise independence. Let be a double sequence of pairwise negatively dependent random variables. If for all non-negative real numbers t and , for 1 < p < 2, then we prove that (1). In addition, it also converges to 0 in ....
متن کاملAsymptotic Behaviors of the Lorenz Curve for Left Truncated and Dependent Data
The purpose of this paper is to provide some asymptotic results for nonparametric estimator of the Lorenz curve and Lorenz process for the case in which data are assumed to be strong mixing subject to random left truncation. First, we show that nonparametric estimator of the Lorenz curve is uniformly strongly consistent for the associated Lorenz curve. Also, a strong Gaussian approximation for ...
متن کاملA Note on the Strong Law of Large Numbers
Petrov (1996) proved the connection between general moment conditions and the applicability of the strong law of large numbers to a sequence of pairwise independent and identically distributed random variables. This note examines this connection to a sequence of pairwise negative quadrant dependent (NQD) and identically distributed random variables. As a consequence of the main theorem ...
متن کاملAn Extension of the Hardy-littlewood Strong Law
A strong law is established for linear statistics that are weighted sums of a random sample. Using an observation of Cheng (1995a) about the Bernstein and Kolmogorov inequalities, we present an extension to the Hardy-Littlewood strong law under certain moment conditions on the weights and the distribution. As a byproduct, the Marcinkiewicz-Zygmund strong law and the law of the iterated logarith...
متن کامل0 M ay 1 99 9 1 THE LIL FOR CANONICAL U - STATISTICS OF ORDER
Let X, X i , i∈N, be independent identically distributed random variables and let h(x,y)= h(y,x) be a measurable function of two variables. It is shown that the bounded law of the iterated logarithm, lim sup n (n log log n) −1 1≤i<j≤n h(X i ,X j) <∞ a.s., holds if and only if the following three conditions are satisfied: h is canonical for the law of X (that is, Eh(X,y)=0 for almost all y) and ...
متن کامل