A note on invariance principles for v. Mises' statistics
Let \((X_ n)_{n\geq 1}\) (resp. \((Y_ n)_{n\geq 1})\) be a process of \(E_ 1\) (resp. \(E_ 2)\)-valued i.i.d. random variables with distribution \(L(X_ 1)\) (resp. \(L(Y_ 1))\). Suppose that the two processes are independent. Using these processes and a measurable function \(h(x_ 1,...,x_{m_ 1},y_{m_ 1+1},...,y_ m)\) \((x_ i\in E_ 1\) \((i=1,...,m_ 1)\), \(y_ j\in E_ 2\) \((j=m_ 1+1,...,m))\) define the two-sample von Mises statistics \(V_ n(h)\) \((n=(n_ 1,n_ 2))\). Finally, define \(T^ h_ n\) \((n\in N^ 2)\) to be the \(D([0,1]^ 2)\)-valued random element given by \[ T^ h_ n(t_ 1,t_ 2)=(n_ 1+n_ 2)^{(m_ 1+m_ 2)/2}t_ 1^{m_ 1}t_ 2^{m_ 2}V_{([n_ 1t_ 1],[n_ 2t_ 2])}(h). \] In this paper, extending \textit{A. A. Filippova}'s result [Teor. Veroyatn. Primen. 7, 26-60 (1962; Zbl 0118.145); English translation in Theory Probab. Appl. 7, 24-57 (1962)], on weak convergence of von Mises functionals, the authors proved a weak convergence principle for \(\{T^ h_ n\}\). Applications to U- statistics and extensions to contiguity and some weakly dependent processes are also considered.
scientific article; zbMATH DE number 3986382
scientific article