Limit Theorem on the Self-Normalized Range for Weakly and Strongly Dependent Process

Abstract

R is a range and Q a self-normalized range. For certain r.f. X(t), one can select the weight function A(d; Q), so that (in some sense) the d ~ limit of either Q (d)/A (d; Q) or Q (e ~ d)/A (e* d; Q) is nondegenerate; if so, A (d; Q) is the key factor in a new statistical technique, called R/S analysis. The theorems in this paper describe some aspects that have already been founded fully upon theorems (easy to prove but unexpected) concerning weak convergence of certain r.f., while the conjectures relate to other aspects of R/S analysis that still rely, at this stage, upon properties suggested by heuristics and by computer simulation. For iid processes satisfying E X 2 < ~:), or attracted to a stable process of exponent ~, it is shown that A = l /~ independently of a. For processes that are weakly dependent (e.g., Markov or autoregressive) one still has A = v/d. Conversely, whenever A = V ~, the r.f. X (t) will be said to have a finite R/S memory. On the other hand, if X(t) are the finite increments of a proper fractional Brownian m o t i o n defined as the fractional integral of order H 0 . 5 : ~ 0 of ordinary Brownian m o t i o n o n e has A = d n. This X(t) is strongly d e p e n d e n t rather than strongly m i x i n g a n d it can be said to have an infinite memory. Conversely, whenever A#]//d, the r.f. X(t) will be said to have an infinite R/S memory. When A = d n L(d), with H ~ 0.5 and