Convergence of functionals of sums of r.v.s to local times of fractional stable motions

By Pratheepa Jeganathan. 2004

56 citations recorded by the source corpus.

Consider a sequence Xk = ∑j=0∞cjξk−j, k ≥ 1, where cj, j ≥ 0, 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<α≤2. 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 γn−1S[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 n−1βn∑k = 1[nt]f(βn(γn−1Sk+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 n−1βn→0. The only further condition that is assumed on the distribution of ξ1 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.

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/cd72865cdbbb292b12f5a43b14661f47e0355deb

How researchers see this paper

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In Finance

1. Peter C. B. Phillips
2. Joon Y. Park
3. Qiying Wang
4. Qi Li
5. Zongwu Cai
6. Yoosoon Chang
7. Jiti Gao
8. Martin Wagner

In Management Science and Operations Research

1. Peter C. B. Phillips
2. Qi Li
3. Joon Y. Park
4. Qiying Wang
5. Zongwu Cai
6. Yixiao Sun
7. Victor Chernozhukov
8. Jeffrey S. Racine

Citations in the corpus

• REGRESSION ASYMPTOTICS USING MARTINGALE CONVERGENCE METHODS (2008, unknown): “…and Phillips, 1999, 2001; Phillips and Magdalinos, 2007; and references therein!+ Recent attention ~Park and Phillips, 1999, 2001; de Jong, 2002; Jeganathan, 2003, 2004; Pötscher, 2004; Saikkonen and Choi, 2004! has also been given to the limit behavior of certain types of nonlinear functions…”
• Functional-coefficient models for nonstationary time series data (2009, result): “By Lemma 9 of Jeganathan (2004), we have µ̂j(z) = µj(K) ∫ 1 0 φε(Wu(s))ds+ oL2(1).”
• A specification test for nonlinear nonstationary models (2012, background): “…(1994) (and their references for related work) for the particular situation where cnxk,n is a partial sum of i.i.d. random variables, and to Akonom (1993), Phillips and Park (1998), Jeganathan (2004) and de Jong and Wang (2005) for the case where cnxk,n is a partial sum of a linear process.”
• ASYMPTOTIC THEORY FOR LOCAL TIME DENSITY ESTIMATION AND NONPARAMETRIC COINTEGRATING REGRESSION (2009, unknown): “As in Jeganathan (2004), the approach in this paper involves approximating the difference cn n n ∑ k=1 g ( cn xk,n )− cn n n ∑ k=1 ∫ ∞ −∞ g [ cn (xk,n + z ) ] φ(z)dz, for some and where φ(x) = 1√ 2π exp {− (x2/2)}.”
• ASYMPTOTIC THEORY FOR ZERO ENERGY FUNCTIONALS WITH NONPARAMETRIC REGRESSION APPLICATIONS (2010, background): “Statement (a) has been established in Jeganathan (2004) (also see Wang and Phillips, 2009a).”