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On sampling of stationary increments processes

Journal article, 2004

Let $X(t)$, $0\leq t\leq 1$, be a stochastic process with stationary increments. Let $q_1(t)$, $q_2(t)$ and $w(t)$, $t>0$, be positive functions of $t$ which tend to $0$ for $t\to 0$, and define the random variables $$Z_1(t)=\{X(t)-X([t/q_1(\epsilon)]q_1(\epsilon))-\epsilon\}/w(\epsilon),$$ and $$Z_2(t)={|X(t)-X([t/q_2(\epsilon)]q_2(\epsilon))|-\epsilon}/w(\epsilon),\text{ for }\epsilon>0.$$ Under a technical condition of eight parts it is shown that $$H(x)=\lim_{\epsilon\downarrow 0}P\{\sup_{0\leq t\leq 1}Z_i(t)\leq x}, i=1,2,$$ exists for $x\in J$, for some open interval $J$, and $H$ is identified. The functions $q_i$, $w$ and $H$ are themselves defined in the formulation of the conditions in the assumptions of the theorem stating the result. Some of these conditions are shown to be similar to those previously used in extreme value theory. Examples with strictly $\alpha$-stable Levy motions are given. The main result is also shown to hold for a totally skewed linear fractional $\alpha$-stable motion.