An invariance principle for reduced U-statistics

Summary

Let (X _n) _n∈N be a sequence of i.i.d. r.v.′s, letf(x₁...x _m ) be a symmetric function ofm arguments. Forn∈N,n≥m, letC(n) ⊂ {(i ₁,...i _m):1 ≤i ₁...i _m ≤n} with the property that eachi∈{1,...n} is present in the same number of elements ofC(n). Define random elementsW _n in the Skorokhod spaceD[0, 1] by

$$W_n (t) = n^{1/2} (\# C(n))^{ - 1} \sum\nolimits_C {([\operatorname{int} ])} (f(X_{i_1 } ...X_{i_m } ) - Ef(X_1 ,...Xm))$$

for 0≤t≤1,n∈N. We give sufficient conditions for tightness and weak convergence of (W _n) _n∈N.