arXiv subjects
Publications and source records attributed to J. M. P. Albin.
Under a complex technical condition, similar to such used in extreme value theory, we find the rate q(ε)^{-1} at which a stochastic process with stationary increments ξshould be sampled, for the sampled process ξ(\lfloor\cdot /q(ε)\rfloor q(ε)) to deviate from ξby at most ε, with a given probability, asymptotically as ε\downarrow0. The canonical application is to discretization errors in computer simulation of stochastic processes.
Let {X(t)}_{t\ge0} be a locally bounded and infinitely divisible stochastic process, with no Gaussian component, that is self-similar with index H>0. Pick constants γ>H and c>0. Let νbe the Lévy measure on R^{[0,\infty)} of X, and suppose that R(u)\equivν({y\inR^{[0,\infty)}:supt\ge 0y(t)/(1+ct^γ)>u}) is suitably ``heavy tailed'' as u\to\infty (e.g., subexponential with positive decrease). For the ``storage process'' Y(t)\equiv sup_{s\ge t}(X(s)-X(t)-c(s-t)^γ), we show that P{sup_{s\in[0,t(u)]}Y(s)>u}\sim P{Y(\hat t(u))>u} as u\to\infty, when 0\le \hat t(u)\le t(u) do not grow too fast with u [e.g., t(u)=o(u^{1/γ})].