Consider Zft (u) = Rtu 0 f(Ns) ds, t > 0, u 2 [0, 1] where N = (Nt)t2R is a normal process and f is a measurable real-valued function satisfying Ef(N0)2 <1 and Ef(N0) = 0. If the dependence is sufficiently weak Hariz  showed that Zf t /t1/2 converges in distribution to a multiple of standard Brownian motion as t 1. If the dependence is sufficiently strong then Zt/(EZt(1)2)1/2 converges in distribution to a higher order Hermite process as t 1 by a result by Taqqu . When passing from weak to strong dependence, a unique situation encompassed by neither results is encountered. In this paper, we investigate this situation in details and show that the limiting process is still a Brownian motion, but a nonstandard norming is required. We apply our result to some functionals of fractional Brownian motion which arise in time series. For all Hurst indices H 2 (0, 1), we give their limiting distributions. In this context, we show that the known results are only applicable to H <3/4 and H > 3/4, respectively, whereas our result covers H = 3/4.
|Pages (from-to)||49 - 70|
|Number of pages||22|
|Journal||Annals of Applied Probability|
|Publication status||Published - 2009|