# Approximation Concepts and Limit Theorems

• Benedikt M. Pötscher
• Ingmar R. Prucha
Chapter

## Abstract

In this chapter we provide formalizations of the notion that a stochastic process has a “fading memory”. Some of these formalizations employ concepts of approximation of one process by another process. The aim of these formalizations is to define classes of processes that — while still satisfying limit theorems (LLNs and CLTs) — are broad and cover, in particular, processes that are generated from a dynamic system.1 In Section 6.1 we start with a discussion of the limitations of the concept of α-mixing [Ø-mixing], followed by the definition of L p -approximability of a stochastic process in Section 6.2. This approximation concept was introduced in Pötscher and Prucha (1991a). It encompasses the approximation concept of stochastic stability and near epoch dependence, and helps to clarify the relationship between these concepts. In Section 6.3 we then discuss LLNs for L p -approximable and near epoch dependent processes. (The discussion of CLTs is deferred to Chapter 10.) Frequently we are interested in limit theorems for a function of an L p -approximable [near epoch dependent] process. E.g., when proving consistency via the use of a ULLN, we need to establish local LLNs, i.e., LLNs for the “bracketing” functions $$q_t^*\left( {{z_t},\theta ;\eta } \right)and{q_{{t_*}}}\left( {{z_t},\theta ;\eta } \right)$$. If the underlying process (z t ) is L p approximable [near epoch dependent] this can be accomplished by making use of results that show under which circumstances functions preserve the L P approximability [near epoch dependence] property. Preservation results of this type are the subject of Section 6.4. In considering dynamic systems it is important to know when the process generated by the system will satisfy the L P approximability [near epoch dependence] property. Hence, in Section 6.4 we also provide sufficient conditions for dynamic systems under which the output process is L p approximable [near epoch dependent]. Since limit theorems for Lp approximable [near epoch dependent] processes are available (cf. Section 6.3 and Chapter 10), such results are fundamental for the derivation of limit theorems for (functions of) processes that are generated by dynamic systems. Several of these results, and in particular those that cover higher order systems, are new and have not been available in the literature previously. Finally, in Section 6.5 we utilize the results developed in this chapter to give sets of sufficient conditions which ensure that q t (z t , θ) satisfies a local LLN, i.e., we provide sufficient conditions for Assumption 5.2 in Chapter 5.

## Keywords

Limit Theorem Contraction Condition Moment Condition Basis Process Borel Subset
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## Reference

1. 1.
For simplicity of presentation the results in this chapter are only presented for sequences of random variables. However, they can be readily extended to triangular arrays.Google Scholar
2. 2.
In order to apply this definition to processes (6)tEN we can put 6 equal to some constant for s O.Google Scholar
3. 3.
A reason for considering the subclass of 0-mixing processes separately is that limit theorems under the assumption of a 7-mixing process can sometimes be derived under weaker additional conditions than is possible under the assumption of an a-mixing process; cf. Theorem 6.4 for example.Google Scholar
4. 4.
For conditions under which linear processes are a-mixing [0-mixing] see, e.g., Ibragimov and Linnik (1971), Chanda (1974), Gorodetskii (1977), Withers (1981a), Pham and Tran (1985), and Mokkadem (1986).Google Scholar