By Kung-Sik Chan, Howell Tong

It used to be none except Henri Poincare who on the flip of the final century, regarded that initial-value sensitivity is a primary resource of random ness. For statisticians operating in the conventional statistical framework, the duty of significantly assimilating randomness generated through a in simple terms de terministic process, often called chaos, is an highbrow problem. Like another statisticians, now we have taken up this problem and our interest as journalists and individuals has led us to enquire past the sooner discoveries within the box. past statistical paintings within the zone was once regularly con cerned with the estimation of what's occasionally imprecisely known as the fractal measurement. through the varied phases of our writing, tremendous parts of the e-book have been utilized in lectures and seminars. those contain the DMV (German Mathematical Society) Seminar software, the inaugural consultation of lectures to the obstacle issues venture on the Peter Wall Institute of complex Stud ies, college of British Columbia and the graduate classes on Time sequence research on the collage of Iowa, the collage of Hong Kong, the Lon don university of Economics and Political technology, and the chinese language collage of Hong Kong. we've as a result benefitted enormously from the reviews and recommendations of those audiences in addition to from colleagues and pals. we're thankful to them for his or her contributions. Our distinct thank you visit Colleen Cutler, Cees Diks, Barbel FinkensHidt, Cindy Greenwood, Masakazu Shi mada, Floris Takens and Qiwei Yao.

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**Example text**

For an interesting discussion of this phenomenon, see the last paragraph on p. 41 in Falconer (1990). (8) We should mention that the verification of assumption (AI) and (A2) need not be easy. In fact, the nature of the attractors of many deterministic maps, for example the Henon map, is still unknown. 38 3. 4) Zero is a globally asymptotically stable fixed point if i¢i < 1. So, G can be chosen to be the whole real line. Subject to the regularity conditions (A4) and (AS), the process is then ergodic, and the support of the noise could be bounded or unbounded.

S. 7 Consider the classical linear ARMA(p, q) model (without loss of generality, it is assumed that q = p - 1) Yt - tPl Yt-l - ... - tPpYt-p = et + (Jlct-l + ... + (Jqct-q, where et satisfies the usual conditions of being iid and independent of past Y s. A state space form can be written as follows: Xt Yt = = FXt - 1 + Get (0,0, ... ,0, I)Xt, 40 3. Chaos and Stochastic Systems where F=I 0 0 1 0 O 1 0 rPp 0 rPp-1 0 rPp-2 0 0 1 rP1 and G = {(Jq,(Jq_1, ••• ,(h, I)T. The spectral radius of F equals the maximum magnitude of the eigenvalues of F.

Let X and X' be two independent 24 2. 2. The periodogram of the first component of 10,000 Henon iterates starting from the origin. random vectors with the natural measure 1f as their common distribution. Let IX - X'leo denote the maximum norm between X and X' which is the maximum of the absolute magnitude of the components of X - X'. The distribution function of IX - X'leo is called the population correlation integral and equals C(r) = P(lX - X'leo ~ r). e. C(r) '" r P , as r -t 0. ) The exponent p is defined as the correlation dimension of the attractor (or more correctly, the natural measure on the attractor).