Abstract
The celebrated results of Komlós, Major and Tusnády [Z. Wahrsch. Verw. Gebiete 32 (1975) 111–131; Z. Wahrsch. Verw. Gebiete 34 (1976) 33–58] give optimal Wiener approximation for the partial sums of i.i.d. random variables and provide a powerful tool in probability and statistics. In this paper we extend KMT approximation for a large class of dependent stationary processes, solving a long standing open problem in probability theory. Under the framework of stationary causal processes and functional dependence measures of Wu [Proc. Natl. Acad. Sci. USA 102 (2005) 14150–14154], we show that, under natural moment conditions, the partial sum processes can be approximated by Wiener process with an optimal rate. Our dependence conditions are mild and easily verifiable. The results are applied to ergodic sums, as well as to nonlinear time series and Volterra processes, an important class of nonlinear processes.
Highlights
Let X1, X2, . . . be independent, identically distributed random variables with EX1 = 0, EX12 = 1
The purpose of the present paper is to develop a new approximation technique enabling us to prove the KMT approximation (1.1) for all p > 2 and for a large class of dependent sequences
In this paper we introduce a new, triadic decomposition scheme enabling one to deduce directly, under the dependence measure (1.5) below, the asymptotic properties of Xn in (1.4) from those of the εn
Summary
KOMLÓS–MAJOR–TUSNÁDY APPROXIMATION and others and led to the theory of weak convergence of probability measures on metric spaces; see, for example, Billingsley (1968) In another direction, Strassen (1964) used the Skorohod representation theorem to get an almost sure approximation of partial sums of i.i.d. random variables by Wiener process. In this paper we introduce a new, triadic decomposition scheme enabling one to deduce directly, under the dependence measure (1.5) below, the asymptotic properties of Xn in (1.4) from those of the εn This allows us to carry over KMT approximation from the partial sums of the εn to those of Xn. To state our weak dependence assumptions on the process in (1.4), assume Xi ∈ Lp, p > 2, namely Xi p := [E(|Xi|p)]1/p < ∞. The following theorem, which is the main result of our paper, provides optimal KMT approximation for processes (1.4) under suitable assumptions on the functional dependence measure.
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