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Article Dans Une Revue Lithuanian Mathematical Journal Année : 2007

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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Dates et versions

hal-00141662 , version 1 (03-05-2007)

Identifiants

Citer

Paul Doukhan, Gabriel Lang, Donatas Surgailis. Limit theorems for sums of non linear function of ARFIMA processes with random Hurst exponents and Gaussian innovations. Lithuanian Mathematical Journal, 2007, 47 (1), pp.1-23. ⟨10.1007/s10986-007-0001-2⟩. ⟨hal-00141662⟩
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