Weak convergence to the multiple Stratonovich integral

We have considered the problem of the weak convergence, as ε tends to zero, of the multiple integral processes∫0t⋯∫0tf(t1,…,tn) dηε(t1)⋯dηε(tn),t∈[0,T]in the space C0([0,T]), where f∈L2([0,T]n) is a given function, and {ηε(t)}ε>0 is a family of stochastic processes with absolutely continuous paths that converges weakly to the Brownian motion. In view of the known results when n⩾2 and f(t1,…,tn)=1{t1<t2<⋯<tn}, we cannot expect that these multiple integrals converge to the multiple Itô–Wiener integral of f, because the quadratic variations of the ηε are null. We have obtained the existence of the limit for any {ηε}, when f is given by a multimeasure, and under some conditions on {ηε} when f is a continuous function and when f(t1,…,tn)=f1(t1)⋯fn(tn)1{t1<t2<⋯<tn}, with fi∈L2([0,T]) for any i=1,…,n. In all these cases the limit process is the multiple Stratonovich integral of the function f.