Holderian invariance principle for linear processes

Let (xi(n))(n >= 1) be the polygonal partial sums process built on the linear processes X(n) = Sigma(i >= 0) a(i)is an element of(n-i), where (is an element of(i))(i is an element of Z) are i.i.d., centered and square integrable random variables with Sigma(i >= 0) a(i)(2) <infinity. We investigate functional central limit theorem for xi(n) in the Holder spaces H(alpha)(o)[0, 1] of functions x : [0, 1] -> R such that vertical bar x(t + h) - x(t)vertical bar = o(h(alpha)) uniformly in t. When Sigma(i >= 0) vertical bar a(i)vertical bar < infinity (short memory case), we show that n(-1/2)xi(n) converges weakly in H(alpha)(o)[0, 1] to some Brownian motion under the optimal assumption that P(vertical bar epsilon(0)vertical bar) >= t) = o(t(-p)), where 1/p = 1/2 - alpha. This extends the Lamperti invariance principle for i.i.d. X(n)'s. When a(i) = l(i)i(-beta), 1/2 < beta < 1, with l positive, increasing and slowly varying, (X(n))(n >= 1) has long memory. The limiting process for xi(n) is then the fractional Brownian motion W(H) with Hurst index H = 3/2 - beta and the normalizing constants are b(n) = c(beta)n(H)l(n). For 0 < alpha < H - 1/2, the weak convergence of b(n)(-1)xi(n) to W(H) in H(alpha)(o)[0, 1] is obtained under the mild assumption that Ec(0)(2) < infinity. For H -1/2 < alpha < H, the same convergence is obtained under P(vertical bar c(0)vertical bar) >= t) = o(t(-p)), where 1/p = H - alpha.