Local Properties of the Autoregressive Series

Abstract

Let us have a normal stationary autoregressive series $\{X_t\}^\infty_{-\infty}$ of the $n$th order with $EX_t = 0$. Denote $b$ the vector of autoregressive parameters. In this paper the Radon-Nikodym derivative $dP_b/dP$ is studied, where $P_b$ is the probability measure corresponding to the finite part (of length $N$) of the autoregressive series and $P = P_0$, i.e., $P$ corresponds to the case, when $X_t$ are independent normal random variables. The function $dP_b/dP$ may be expanded in the power series of components of vector $b$. If the norm $\|b\|$ is small, then the absolute term and the linear terms are most important. These terms are given in the paper and they are used for an approximation of the probability $P_b(A)$, where $A$ is a Borel set in the $N$-dimensional Euclidean space $R_N$. The probability that a normal stationary autoregressive series does not exceed a constant barrier is analysed as an example. A second example is devoted to the properties of the sign-test when the observations are dependent and may be described by the autoregressive model.