On Minimality and Interpolation of Harmonizable Stable Processes

On basicity of exponentials inL P (dμ) and general prediction problems

“…Using the results of section 3 in conjunction with the General prediction Theorem 2.5, one can furnish new proofs for the well-known prediction theorems of Szeg6, Kolmogorov and Yaglom, as well as their recent generalizations to non-second-order processes given in [1], [6], [7], [8], [12]. The proofs are not just shorter, but give a uniform approach to all these and other prediction results.…”

“…In the last Theorem, taking n = § one can immediately prove Kolmogorov's prediction Theorem for the case p = 2 as well as its extension to the general case of 1 < p < oo due to eourahmadi [8] and Weron [12]. Taking n = 0, one has Szeg6's prediction error formula for the case p = 2 and its extension to the non-second order case (cf.…”

“…These types of M's play important rules in the prediction theory of stochastic processes. In Section 4 we apply results of Section 3 to get some general prediction theorem which subsume the classical prediction results of Szeg6, Kolmogorov, Yaglom, and their recent generalization for the p-th order processes given in [1,7,8,12]. The main tools we will need are Buerling's Theorem about outer functions, a general prediction theorem proved in [5,Theorem 3.1], and a new representation of the dual of LP(w) as L~(w') instead of its traditional representation Lq(w).…”

On Prediction of Harmonizable Stable Processes.

“…The general multi-step case was considered by Cambanis and Soltani (1984). The problem of interpolation has been considered by Pourahmadi (1984) and also by Weron (1985) in a more general set-up along with some ergodic properties.…”

“…For the second order processes this question has been the subject cf study by several authors: Akutowicz (1957), Masani (1960), Miamee ond Salehi (1923), Pourahmadi (1984Pourahmadi ( ),(1985, and Bloomfield (1984). Some such conditions are niven in 4 Proposition 13, which is inspired by the conditions in Bloomfield (1984) …”