Spectral analysis of the covariance of the almost periodically correlated processes

Discrete Periodic Sampling with Jitter and Almost Periodically Correlated Processes

“…Whenever the process X is stationary then all the spectral density functions f (λ, ·), λ ∈ R except for λ = 0, are identically null (Dehay, 1994), and we find Akaike's results (1960, p. 153) as a particular case of the previous equalities. Moreover, for an UAPC process we see that the timing perturbations act as filters on any of the components of the folding decomposition (8) of the spectral covariance of the observed process.…”

“…If, in addition, the function α → λ∈ f (λ, α) is integrable on R, that is, if the process X is strongly harmonizable (Dehay, 1994), then we can readily state that…”

“…Whenever the process X is ρ-mixing (for the definition see Section 2.3) with mixing function ρ( · ) ∈ L 1 (R + ), then it has spectral density functions f (λ, ·), λ ∈ , see Dehay (1994) and Hurd (1991). Indeed for any λ ∈ , the function τ → a(λ, τ ) is integrable on R, the spectral density function f (λ, ·) is defined by…”

Weak law of large numbers for almost periodically correlated processes

“…We will say that a process X satisfies the weak law of large numbers (WLLN) if the limit (2) exists in the norm topology of H.…”

“…An APC process X is called almost periodically unitary (APU) if there is a continuous unitary group U (t), t ∈ R, and an H-valued uniformly almost periodic function f such that X(t) = U(t)f(t), t ∈ R [5]. An APC process X is called uniformly almost periodically correlated (UAPC) if the function u −→ B(·, u) is uniformly almost periodic from R to C(R), the Banach space of bounded continuous functions on R equipped with the sup-norm [2].…”