Abstract. Consider an i.i.d. sequence $\{\zeta_i\} _{i=-\infty }^\infty $, a sequence of real numbers $\{b_i\} _{i\geq0}$ and the pertaining infinite-order moving average $\epsilon_i=\sum_{j\leq i}b_{i-j}\zeta_j$, $i=1,2,\ldots $. Under conditions on $\{b_i\} $ which entail that $\{\epsilon_i\} $ is either long-range or short-range dependent, we study the partial-sum process $S_n(t)=\sum_{i=1}^{\lfloor nt\rfloor }\epsilon_i$, $t\geq0$. For $0< b< \infty $, $k\in{\msbm N}$, a suitable norming sequence $\{a_n\} $ and sequences of gap-lengths $l_{1,n}, \ldots, l_{k,n}$ such that $l_{1,n}\to\infty $ and $l_{j,n} - l_{j-1,n}\to\infty $, $j = 2, \ldots, k$, we prove in the first case that the vector process $a_n(S_n(t_0), S_n(l_{1,n}+t_1)-S_n(l_{1,n}), \ldots, S_n(l_{k,n}+t_k)-S_n(l_{k,n}))$, $0\leq t_0,\ldots,t_k\leq b$, converges in distribution in ${\cal D}[0,b]^{k+1}$ to the vector of ${k+1}$ independent fractional Brownian motions. The result is then generalized to the case when $\{\epsilon_i\} $ is replaced by values of an $m$-th Appell polynomial $\{P_m(\epsilon_i)\},m\in{\msbm N}.$ In the short-range dependence case $$n^{-1/2}(S_n(t_0), S_n(l_{1,n}+t_1)-S_n(l_{1,n}), \ldots, S_n(l_{k,n}+t_k)-S_n(l_{k,n}))$$ converges in distribution to the vector of $k+1$ independent Wiener processes, provided $l_{j,n}-l_{j-1,n}\geq b+\gamma $ for some $\gamma >0$, $j=1,\ldots,k$. As applications, in both cases we determine the asymptotic behaviour of the finite-dimensional distributions of kernel estimators in the fixed-design regression model with errors $\epsilon_i$. The results parallel those of Csörgő and Mielniczuk [3], [4] when $\{\epsilon_i\} $ is an instantaneous transformation of a Gaussian sequence.

AMS Subject Classification (1991): 60F05, 62G07

Received December 10, 1996 and in revised form December 30, 1996. (Registered under 6117/2009.)