• Peter Bloomfield
• William L. Steiger
Part of the Progress in Probability and Statistics book series (PRPR, volume 6)

## Abstract

This chapter is devoted to stationary, kth order autoregressions
$${X_n} = {a_0} + {a_1}{X_{n - 1}} + \ldots + {a_k}{X_{n - k}} + {U_n}$$
(1)
where...,U-1,U0,U1,... is an i.i.d. sequence of random variables. The first question that arises is: what conditions on a and the Ui will assure that there actually exists a stationary sequence {Xi} satisfying (1)? Using (1) recursively for Xn-1 in (1), then again for Xn-2, etc., one obtains after N substitutions,
$${X_n} = d(N) + \sum\limits_{j = 0}^N {{b_j}{U_{n - 1}}} + \sum\limits_{j = N + 1}^{N + k} {{c_j}(N){X_{n - j}}}$$
(2)
for a certain unique choice of bj,cj(N); i.e., b0 = 1, b1 = a1, etc.

## Keywords

Error Distribution Stationary Sequence Ergodic Theorem Stationarity Condition Double Exponential

## Notes

1. 1.
in the classical case of (1.1), E(Ui 2) < ∞ is assumed [this certainly implies (1.5)]. With stationarity, (3), Σ bjUn-j converges in L2, therefore in probability. Kanter and Steigert 1974) gave the first generalization to the case where Ui is attracted to a non-normal stable law of index α. The condition (1.5) and Lemma 1.1 is from Yohai and Maronna (1977). Being in dom(a) is much more restrictive than the Yohai-Maronna condition, (1.5). However they impose, though don’t use, symmetry of Ui.Google Scholar
2. 2.
The theorem in (1.9) began in Kanter and Steiger (1974) with ĉa in probability. At the 1974 Brasov Conference [Kanter and Steiger (1977)] they noticed that their consistency proof actually implied the rate Nδ (ĉ N-a) → 0 in probability, δ < min(1/α,(2-α)/α). Kanter and Hannan (1977) cleaned up the bound on 6 and established convergence with probability 1. Under a weaker condition than Ui ∈ Dom(α), namely (1.5), Yohai and Maronna established a weaker result, namely that Nδ(ĉ N-a) → 0 in probability, δ < 1/2. Thus a wider class of processes is embraced at the expense of weakening the convergence rate. Again, we cite Feller (1971) as a convenient source for details about stable laws.Google Scholar
3. 3.
Theorem 2.1 is from Gross and Steiger (1979). The statement of Theorem 2.2 was one of two conjectures they made.Google Scholar
4. 4.
Theorems 2.2 and 2.3 are from An and Chen (1982). The proof that N-tS2 → ∞ in (1.12) is similar to theirs. Our proof that N-tS1 → 0 in (1.11) is much simpler. Curiously, they give an example, which, if correct, would seem to invalidate their method of proof of Theorem 2.2.Google Scholar
5. 5.
All that is needed for Theorem 2.3 is that Np times the expression in (2.29) converge to zero. The original proof of this is more complicated than ours.Google Scholar