The purpose of this paper is to give a unified treatment of a number of almost sure convergence theorems by exploiting the fact that the processes involved possess a common “almost supermartingale” properties To be precise, let (Ω,F,P) be a probability space and F1 ⊂ F2 ⊂ … a sequence of sub-σ-algebras of F. For each n= 1,2, … let zn, βn, ξn, and ζn be non-negative Fn -measurable random variables such that 

$$E({z_{n + 1}}|{F_n}) \leqslant {z_n}(1 + {\beta _n}) + {\xi _n} - {\zeta _n}.$$

(1)

.

A Convergence Theorem for Non Negative Almost Supermartingales and Some Applications

We suggest a sequential monitoring scheme to detect changes in the parameters of a GARCH(p,q) sequence. The procedure is based on quasi-likelihood scores and does not use model residuals. Unlike for linear regression models, the squared residuals of nonlinear time series models such as generalized autoregressive conditional heteroskedasticity (GARCH) do not satisfy a functional central limit theorem with a Wiener process as a limit, so its boundary crossing probabilities cannot be used. Our procedure nevertheless has an asymptotically controlled size, and, moreover, the conditions on the boundary function are very simple; it can be chosen as a constant. We establish the asymptotic properties of our monitoring scheme under both the null of no change in parameters and the alternative of a change in parameters and investigate its finite-sample behavior by means of a small simulation study.This research was partially supported by NSF grant INT-0223262 and NATO grant PST.CLG.977607. The work of the first author was supported by the Hungarian National Foundation for Scientific Research, grants T 29621, 37886; the work of the second author was supported by NSERC Canada.

SEQUENTIAL CHANGE-POINT DETECTION IN GARCH(p,q) MODELS

A CONVERGENCE THEOREM FOR NON NEGATIVE ALMOST SUPERMARTINGALES AND SOME APPLICATIONS**Research supported by NIH Grant 5-R01-GM-16895-03 and ONR Grant N00014-67-A-0108-0018.

Change-point in the mean of dependent observations

On the detection of changes in autoregressive time series I. Asymptotics

The original Kolmogorov's inequality [6] has been extended to a martingale inequality by Levy [8] and Ville [12] and later to a semimartingale inequality by Doob [3]. In this note we will extend (1) to a semi-martingale inequality which contains Doob's inequality as a special case. As Kolmogorov's inequality is the key to the proof of the law of large numbers for a sequence of independent random variables, we will use our inequality to prove a "law of large numbers" for a martingale, which will be shown to include the extensions of Kolmogorov's law of large numbers for independent random variables [7] made by Brunk [I], Chung [2], Kawata and Udagawa [5], and Prohorov [11], and for dependent random variables made by Levy [8] and Loeve [9]. In the following (W, F, P) will be a probability space, cl, c2, . . . a nonincreasing sequence of positive numbers, xl, x2, * * * a sequence of random variables, yk=XlX2+x2 ? * +Xk and Fk the Borel field generated by xi, x2, * * *, Xk for each k, and for a random variable z we put z+=max(z, 0).

/pdf/a-martingale-inequality-and-the-law-of-large-numbers-9omgcot3jj.pdf

Y. S. Chow

Papers

A martingale inequality and the law of large numbers