Some remarks on duality of stationary sequences

Abstract

Abstract. The paper clarifies the connection between Urbanik's and Miamee and Pourahmadi's concepts of duality for univariate weakly stationary random sequences. Some of Urbanik's results are proved in an alternative way and at the same time generalized to the multivariate case.

Duals of random vectors and processes with applications to prediction problems with missing values

“…Concerning the question above, it is worth noting that Nakazi's technique, if viewed properly, reduces the computation of σ 2 (f , S n ) to that of the (n + 1)-step prediction error variance of another stationary process {Y t } with the spectral density function f −1 , which we call the dual of {X t } (see Definition 1 and Section 4.3). His result and technique have spawned considerable research in this area in the last two decades; see Miamee and Pourahmadi (1988), Miamee (1993), Cheng et al (1998), Frank and Klotz (2002), Klotz and Riedel (2002) and Bondon (2002). A unifying feature of most of the known results thus far seems to be a fundamental duality principle of the form…”

“…As a consequence, we find a new and explicit formula for the dual of the random process {X t ; t ≤ n} for a fixed n, which does not seem to be possible using the technique of Urbanik (2000), Klotz and Riedel (2002) and Frank and Klotz (2002). This is particularly useful in developing series representations for predictors and interpolators, and sheds light on the approaches of Bondon (2002) and Salehi (1979).…”

A prediction problem in $L^2 (w)$

“…Neither problem is particularly simple but the latter seems simpler. In [2,Theorems 5,6], an orthogonalization method is used to compute σ 2 (w, S 3 ). Then the duality relation (2.4) is used to give σ 2 (w, S 2 ), yielding…”

“…, n}, n ≥ 1, has generated considerable interest in computing σ p (w, S) when the index set S is S 0 with finitely many points of Z added or deleted. To name some related contributions, let us mention here Cheng et al [2], Frank and Klotz [4], Klotz and Riedel [6], Kolmogorov [7], Miamee and Pourahmadi [9], Pourahmadi [13], [14], and Urbanik [15]. At present, the best known general result is Theorem 2 of Cheng et al [2], which states that, for such an S, σ p (w, S) is positive if and only if log w ∈ L 1 (dµ).…”

Applications of a finite-dimensional duality principle to some prediction problems

“…In this regard, it is worth noting that Nakazi's technique, if interpreted properly, amounts to reducing computation of σ 2 (f, S n ) to that of the (n + 1)-step prediction error variance of another stationary process {Y t } with the spectral density function f −1 which turns out to be the dual of {X t } (see Definition 2.1 and Section 3.5). His result and technique have spawned considerable research in this area in the last two decades; see Miamee and Pourahmadi (1988), Miamee (1993), Cheng et al (1998), Frank and Klotz (2002), Klotz and Riedel (2002) and Bondon (2002). A unifying feature of most of the known results thus far seems to be a fundamental duality principle (Cheng et al (1998), Urbanik (2000)) of the form…”