Weak invariance principles for weighted \(U\)-statistics
This paper deals with the weak limit behaviour of weighted, degenerate \(U\)-statistics \(U_ n=\sum_ n'a_ Ih(X_ I)\), where \(\sum_ n'\) denotes the sum over all multi-indices \(I=(i_ 1,\dots,i_ m)\in\{1,\dots,n\}^ m\), \((a_ I)\) is a symmetric array of real numbers and \(X_ I=(X_{i_ 1},\dots,X_{i_ m})\), where \(X_ 1,X_ 2,\dots\) is a sequence of independent and identically distributed (i.i.d.) random variables. The distribution of \(U_ n\) is shown to be close, in the Lévy-Prokhorov metric, to the distribution of a certain function \(V_ n\), in i.i.d. normal random variables. For the case \(m\geq 2\) a Rotar'-type invariance principle for degenerate \(U\)-statistics is established.
scientific article; zbMATH DE number 501135
scientific article