Convergence of functionals of sums of RVs to local times of fractional stable motions

• 成果类型: Article
• 署名作者: Jeganathan, P
• 署名单位: Indian Statistical Institute; Indian Statistical Institute Bangalore
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117904000000658
• 发表日期: 2004
• 页码: 1771-1795
• 关键词: series

摘要

Consider a sequence X-k = Sigma(j=0) (infinity) c(j)xi(k-j) k greater than or equal to 1, where c(j), j greater than or equal to 0, is a sequence of constants and xi(j), -infinity < j < infinity, 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 < alpha < 2. Let S-k = Sigma(j=1)(k) X-j. Under suitable conditions on the constants c(j) it is known that for a suitable normalizing constant gamma(n), the partial sum process gamma(n)(-1)S([nt]) converges in distribution to a linear fractional stable motion (indexed by alpha and 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) beta(n)Sigma(k=1)([nt])S(k)f(beta(n)(gamma(n)(-1)S(k) + x)) converges in distribution to (integral(-infinity)(infinity)((.)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 P --> infinity such that n(-1) beta(n) --> 0. The only further condition that is assumed on the distribution of xi(1) is that either it satisfies the Cramer's condition or has a nonzero absolutely continuous component. The results have motivation in large sample inference for certain nonlinear time series models.