Download Analysis of Financial Time Series (Wiley Series in by Ruey S. Tsay PDF

By Ruey S. Tsay

ISBN-10: 0470644559

ISBN-13: 9780470644553

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Additional info for Analysis of Financial Time Series (Wiley Series in Probability and Statistics - Applied Probability and Statistics Section Series)

Example text

Consider algorithm (41). 3, 48 M(z, h) = {(z + Ah) E T 1 c(z + Ah) = min{c(z + Ah)/ A 3 0, (z + X’h) E T for all A’ E 10, A]}}. 24 1 PRELIMINARY RESULTS If (zi} is a sequence constructed by algorithm (41), then either {zi} is finite and its last element is desirable, or {zi}is infinite and every accumulation point of {zi} is desirable. Proof. , that it cannot jam up at a point z k , while constructing an infinite sequence of ~ , and at the same time finding that # ( B ~ E ~,zk) -~ < vectors hj E H ( B ~ E , +zk)* - - a B j ~ ~ -Suppose ~.

The models that we shall describe in this section will enable us to establish the principles of our approach to adaptive truncation of infinite subprocedures. In Appendix A we shall present an open-loop approach to truncation. While the idea of conceptual and implementable algorithms is quite subjective, the reader will find this concept extremely useful in deciding whether it does or does not make sense to add the special truncation procedures that we shall describe to a specific algorithm that he wishes to program.

Consider any one of the algorithm models in this section and suppose that it satisfies the assumptions of the corresponding convergence theorem. In addition, suppose that c(z) is bounded from below for all z E T and that if z’ # z” are two desirable points in T, then c(z’) # c(z”). Let {z~}:=~be an infinite sequence in T constructed by this algorithm model. If either T or C’(zo) = { z E TI c(z) < c(zJ} is compact, then zi+ P as i 03, where P is a desirable point. - Proof. Since the sequence {zi} is compact, it must have accumulation points which, as we have already shown, must be desirable.

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