Limit theorems for sums of non linear function of ARFIMA processes with random Hurst exponents and Gaussian innovations
Résumé
Philippe et al. (2005, 2006) introduced time-varying mutually invertable fractionally integrated filters $ A({\bf d}), B({\bf d}) $ depending on an arbitrary sequence ${\bf d} = (d_t, t \in {\Z}) $ of real numbers, and such that, for constant $d_t \equiv d $, $ A({\bf d})=B({\bf d}) = (1 - L)^{-d}$ is the usual fractional integration operator. They also studied partial sums limits of filtered white noise nonstationary processes $A({\bf d}) \veps_t$ and $B({\bf d}) \veps_t $, for certain classes of deterministic sequences ${\bf d}$. The present paper discusses the randomly fractionally integrated stationary processes $X^A_t = A({\bf d}) \veps_t$ and $X^B_t = B({\bf d}) \veps_t$ by assuming that ${\bf d} = (d_t, t \in {\Z}) $ is a random iid sequence, independent of the noise $(\veps_t)$. In the case when the mean $\bar d = \E d_0 \in (0, 1/2)$, we show that large sample properties of $X^A $ and $X^B$ are similar to FARIMA($0,\bar d,0$) process; in particular, their partial sums converge to a fractional Brownian motion with parameter $\bar d +(1/2)$. The main part of the paper is given to the study and characterization of limit distributions of partial sums of nonlinear functions $h(X^A_t)$ of randomly fractionally integrated process $X^A_t$ with Gaussian noise. We show that that the limit distribution of sums of $h(X^A_t)$ is determined by certain integer defined via conditional Hermite expansion of $h$.
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