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July 2004 Convergence of functionals of sums of r.v.s to local times of fractional stable motions
P. Jeganathan
Ann. Probab. 32(3): 1771-1795 (July 2004). DOI: 10.1214/009117904000000658

Abstract

Consider a sequence Xk=j=0cjξkj, k1, where cj, j0, is a sequence of constants and ξj, <j<, is a sequence of independent identically distributed (i.i.d.) random variables (r.v.s) belonging to the domain of attraction of a strictly stable law with index 0<α2. Let Sk=j=1kXj. Under suitable conditions on the constants cj it is known that for a suitable normalizing constant γn, the partial sum process γn1S[nt] converges in distribution to a linear fractional stable motion (indexed by α and H, 0<H<1). A fractional ARIMA process with possibly heavy tailed innovations is a special case of the process Xk. In this paper it is established that the process n1βnk=1[nt]f(βn(γn1Sk+x)) converges in distribution to ( f( y) dy)L(t,x), where L(t,x) is the local time of the linear fractional stable motion, for a wide class of functions f( y) that includes the indicator functions of bounded intervals of the real line. Here βn such that n1βn0. The only further condition that is assumed on the distribution of ξ1 is that either it satisfies the Cramér’s condition or has a nonzero absolutely continuous component. The results have motivation in large sample inference for certain nonlinear time series models.

Citation

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P. Jeganathan. "Convergence of functionals of sums of r.v.s to local times of fractional stable motions." Ann. Probab. 32 (3) 1771 - 1795, July 2004. https://doi.org/10.1214/009117904000000658

Information

Published: July 2004
First available in Project Euclid: 14 July 2004

zbMATH: 1049.60019
MathSciNet: MR2073177
Digital Object Identifier: 10.1214/009117904000000658

Subjects:
Primary: 60F05 , 60G18 , 60J55
Secondary: 62J02 , 62M10

Keywords: fractional ARIMA process , fractional Brownian motion , Fractional stable motion , functionals of sums of fractional ARIMA , heavy tailed distributions , Local time , weak convergence to local times

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 3 • July 2004
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