On the Extremal Behavior of Sub-Sampled Solutions of Stochastic Difference Equations

M.G. Scotto and K.F. Turkman

Abstract: Let $\{X_k\}$ be a process satisfying the stochastic difference equation $$ X_k=A_{k}X_{k-1}+B_k, k=1,2,..., $$ where $\{A_k,B_k\}$ are i.i.d. $\bkR^2$-valued random pairs. Let $Y_k=X_{Mk}$ be the sub-sampled series corresponding to a fixed systematic sampling interval $M>1$. In this paper, we look at the extremal properties of $\{Y_k\}$. Motivation comes from the comparison of schemes for monitoring financial and environmental processes. The results are applied to the class of bilinear and ARCH processes.

Keywords: Stochastic difference equation; systematic sampling; extreme values; extremal index; compound Poisson process; ARCH process; bilinear process.

Classification (MSC2000): 60F05, 60G10, 60G70.

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