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Upper and lower bounds for the tail of the invariant distribution of some \(AR(1)\) processes. (English) Zbl 0926.62080
Szyszkowicz, Barbara (ed.), Asymptotic methods in probability and statistics. A volume in honour of Miklós Csörgő. ICAMPS ’97, an international conference at Carleton Univ., Ottawa, Ontario, Canada, July 1997. Amsterdam: North-Holland/ Elsevier. 723-730 (1998).
Let \(\{X_n\}\) be an AR(1) process defined by \(X_{n+1}=\rho X_n +\eta_{n+1}\), where \(0<\rho<1\) and \(0\leq \eta_n\leq 1\) are i.i.d. random variables with a continuous distribution. Assume that the support of this distribution has the right endpoint 1. Then it is proved that the stationary distribution of \(X_n\) is continuous with right endpoint \((1-\rho)^{-1}\). If \(\eta_1\sim U[0,1]\) then a lower bound and an upper bound of the stationary density of \(X_n\) near the right endpoint are derived as well as bounds for the tail of the stationary distribution. The results are extended to some more general AR(1) processes.
For the entire collection see [Zbl 0901.00049].
Reviewer: J.Anděl (Praha)

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60G10 Stationary stochastic processes