Limit Joint Distribution of $$\boldsymbol{U}$$ -Statistics, $$\boldsymbol{M}$$ -Estimates, and Sample Quantiles

Abstract

Let \(X_{1},X_{2},\ldots,X_{n}\) be independent identically distributed random vectors. Consider a vector \(V(X_{1},X_{2},\ldots,X_{n})\) whose each component is either a \(U\)-statistic or an \(M\)-estimator. Sufficient conditions for asymptotic normality of the vector \(V(X_{1},X_{2},\ldots,X_{n})\) are obtained. In the case when \(X_{1},X_{2},\ldots\) are one-dimensional, sufficient conditions for asymptotic normality are obtained for a vector, each component of which is either a \(U\)-statistic, or an \(M\)-estimator, or a sample quantile.