Convergence of functionals of sums of r.v.s to local times of fractional stable motions
| dc.creator | Jeganathan, P. | |
| dc.date | 2004-10-05 | |
| dc.date.accessioned | 2026-07-07T05:12:52Z | |
| dc.date.available | 2026-07-07T05:12:52Z | |
| dc.description | Consider a sequence X_k=\sum_{j=0}^{\infty}c_jξ_{k-j}, k\geq 1, where c_j, j\geq 0, is a sequence of constants and ξ_j, -\infty <j<\infty, 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<α\leq 2. Let S_k=\sum_{j=1}^kX_j. Under suitable conditions on the constants c_j it is known that for a suitable normalizing constant γ_n, the partial sum process γ_n^{-1}S_{[nt]} converges in distribution to a linear fractional stable motion (indexed by αand H, 0<H<1). A fractional ARIMA process with possibly heavy tailed innovations is a special case of the process X_k. In this paper it is established that the process n^{-1}β_n\sum_{k=1}^{[nt]}f(β_n(γ_n^{-1}S_k+x)) converges in distribution to (\int_{-\infty}^{\infty}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 β_n\to \infty such that n^{-1}β_n\to 0. The only further condition that is assumed on the distribution of ξ_1 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. | |
| dc.description | Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000658 | |
| dc.identifier | https://arxiv.org/abs/math/0410097 | |
| dc.identifier | http://arxiv.org/abs/math/0410097 | |
| dc.identifier | Annals of Probability 2004, Vol. 32, No. 3A, 1771-1795 | |
| dc.identifier | doi:10.1214/009117904000000658 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/72743 | |
| dc.subject | Probability | |
| dc.subject | 60F05, 60G18, 60J55 (Primary) 62M10, 62J02. (Secondary) | |
| dc.title | Convergence of functionals of sums of r.v.s to local times of fractional stable motions | |
| dc.type | text |