The weak convergence for self-normalized UU-statistics with dependent samples

Let UnUn be a UU-statistic based on a symmetric kernel h(x,y)h(x,y) and φ∗φ∗-mixing samples {X,Xn;n≥1}{X,Xn;n≥1}. Under some suitable conditions, the weak convergence of the self-normalized UU-statistics is obtained, which extends a previous result for independent cases.