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Asymptotic expansions for central limit theorems for general linear stochastic processes. I: General theorems on rates of convergence
Author(s) -
Butzer P. L.,
Gather U.,
Törnig W.
Publication year - 1979
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670010208
Subject(s) - mathematics , central limit theorem , generalization , mathematical proof , limit (mathematics) , weak convergence , type (biology) , convergence of random variables , convergence (economics) , rate of convergence , order (exchange) , mathematical analysis , pure mathematics , random variable , ecology , channel (broadcasting) , statistics , geometry , computer security , engineering , finance , computer science , electrical engineering , economics , asset (computer security) , biology , economic growth
This paper deals with so‐called general linear stochastic processes (GLSP), defined by T. Kawata in 1972 in generalization of work of R. Lugannani and J. B. Thomas of 1967/71. These second order processes (which are not necessarily stationary nor have independent increments) are described by rather weak requirements, so that several processes such as some random noise and pulse train processes are specific models of these GLSP. Part I is concerned with two general theorems giving asymptotic expansions (including those for the density function) in the central limit theorem for such GLSP, together with error rates. The assumptions for the corresponding θ– and o –error estimates seem rather natural: in the former, apart from assumptions on the inherent structure of such GLSP, the existence of certain moments of higher order as well as a Cramer‐type condition are assumed, in the latter in addition a Lindeberg‐type condition of higher order. Fourier analytic machinery is used for the proofs.