Central Limit Theorems revisited

Characterizations of Multinormality and Corresponding Tests of Fit, Including for Garch Models

“…Since EZ j (t) = 0, t ∈ R d , and, in view of the condition γ > 1, E Z 1 2 L 2 < ∞, a Hilbert space CLT (see [41]) yields A n + B n D −→ W for a centered Gaussian element W of L 2 having covariance kernel C(s, t) = E[Z 1 (s)Z 1 (t)]. Using E cos s ⊤ X e t ⊤ X = e ( t 2 − s 2 )/2 cos s ⊤ t ,…”

Asymptotic distribution for periodograms of infinite dimensional discrete time periodically correlated processes

“…In this section we apply the Vakhania isometry between a complex Hilbert space and its real counterpart Hilbert space to rewrite the CLT for a sequence of independent random elements in H, [5]. Let us first present the Vakhania imbedding of a complex Hilbert space into a real Hilbert space.…”

“…To state the CLT on a complex Hilbert space H, the theorem of Kundua et al [5], will be restated on H R ⊕H R in Theorem 3.1, and then its version for complex Hilbert spaces will be presented in Corollary 3.1. We let {e k ; k ≥ 1} to be an orthonormal basis for the complex Hilbert space H. We denote the corresponding orthonormal basis for H R ⊕ H R by {e j k ; k ≥ 1, j = 1, 2}, where e 1 k = (e k , 0) tr and e 2 k = (0, e k ) tr .…”

“…In Let ξ be a PC H-V[SSO]SP with spectral representation (5). Since for a fixed s ∈ [0, 2π ) the operator V n (s) given by…”

“…This theorem exists in the literature for real Hilbert spaces [5]. But, to the best of our knowledge, the theorem is not produced for complex Hilbert spaces.…”