Thresholds of moving average of stationary processes for given on target significant levels

Abstract

A Markov chain is associated to a finite order one-sided moving average of a discrete time stationary Gaussian process. A method is developed to specify thresholds ${0=L_0 < L_1 < \cdots < L_m < L_{m+1}=\infty}$ for given on target significant levels ${\pi_0, \ldots, \pi_{m},\quad \sum_{i=0}^{m} \pi_{i} =1;}$ in the sense that in the long run the probability that the moving average process lies in [L _i , L _i+1), will be π _i , i = 0,. . . ,m. Special inputs, AR(1) and MA(1) are treated in details. This article extends the work of Soltani et al. in (Commun Stat Theory Methods 36(14):2595–2606) where the inputs were assumed to be i.i.d.; and a single threshold was considered.