A CLT for the periodograms of a ϱ∗-mixing random field

On spectral and bispectral estimator of the parameter of nongaussian data

“…Suppose that conditions of Theorem 1 and Lemma 3 are fulfilled. Then the estimator ϋτ denned by (9) is asymptotically Gaussian ο)) as Τ -> oo, where Σ 0 (ο 0 ) is defined by (14) and σ 2 (ϋ 0 ) -by (17 Theorem 3 follows from Theorem 2, Lemmas and Slutsky's argument. Theorem 2 can be derived from Rosenblatt's version of the central limit theorem see Rosenblatt [22, p. 63] for two dimensional time series.…”

“…The equation (5) is based on the οον(/3τ(λι, λ2),/3τ(^ι, ^2)) · That later one is calculated by the formula of expressing cumulants of some products by cumulants see Brillinger [6], Leonov and Shiryaev [17] and Appendix later. From now on we shall assume that BIT < Β 2 τ· Most of the results listed above are in the monographs Brillinger [6] and Rosenblatt [22] further investigations can be found by Van Ness [27] and Alekseev [1].…”

Central limit theorem for Fourier transform and periodogram of random fields

“…By using Theorem 25.4 in Billingsley (1995), the limit (23) shows that, the proof of Theorem 4 is now reduced to prove the central limit theorem for M n (t).…”

“…Volný (2015) discovered that the fields of stationary orthomartingales require the ergodicity in only one of the directions of the field as a necessary condition for the CLT. It should also be mentioned that a central limit theorem for periodogram of random fields was obtained by Miller (1995) under mixing conditions. All these papers were inspirational.…”

Spectral Density for Third Order Cumulants under Strong Mixing Conditions

“…See the theorems in [4] and [5]. Also, if \*(s) Ä 0 as s Ä , then we can estimate f (*) from data; see Theorem 2 in [9]. \(s), :(s), \*(s), and :*(s) are related in an interesting way.…”