Lévy Moving Averages and Spatial Statistics

in collaboration with Nicky Best and Katja Ickstadt 1 ' & $ % Moving Averages One common and flexible way of constructing stationary time series (discrete-time stochastic processes) is to begin with an i.i.d. X i ≡ j b j ζ i−j where j typically runs from 0 to some finite q ∈ N (or occasionally from 0 to ∞ or even from −∞ to ∞). It is straightforward to compute the mean and covariance of X i and (at least if the polynomial b(z) = q j=0 b j z j has all its roots in the unit disk) to do OLS forecasting. 2 ' & $ % The obvious analog for continuous time would be to construct a stochastic integral X t ≡ t −∞ b(t − s) ζ(ds) for some random measure ζ(ds) which is " i.i.d. " in that it: • assigns independent random variables to disjoint sets, and • assigns the same distribution to all translates ζ(t + B) Such a continuous-time process could be constructed as the limit as ǫ → 0 of time series X t ≡ X i for t = iǫ, ζ i = ζ (iǫ, iǫ + ǫ]. 3 ' & $ % The two requirements above ζ(A) ⊥ ⊥ ζ(B), ζ(t + B) ∼ ζ(B) impose strict limits on the probability distribution that ζ(B) may have: they imply that the stochastic process ζ t ≡ ζ (0, t] must have stationary independent increments, and hence by the Lévy-Khinchine formula it must admit the representation