More on the Convergence of Gaussian Convex Hulls

• Published in: Journal of mathematical sciences (New York, N.Y.) N.Y.), 2024, Vol.286 (5), p.684-691
• Main Authors: Davydov, Yu, Paulauskas, V.
• Format: Article
• Language: English
• Subjects: Banach spaces; Convergence; Convexity; Hulls; Mathematics; Mathematics and Statistics; Metric space; Normal distribution; Probability theory
• ISSN: 1072-3374; 1573-8795
• DOI: 10.1007/s10958-024-07534-x

A “law of large numbers” for consecutive convex hulls of weakly dependent Gaussian sequences { X n } with the same marginal distribution is extended to the case where the sequence { X n } has a weak limit. Let B be a separable Banach space with a conjugate space B ∗ . Let { X n } be a centered B - valued Gaussian sequence satisfying two conditions: (1) X n ⇒ X ; (2) for every x ∗ ∈ B ∗ , lim n , m , n - m → ∞ E ⟨ X n , x ∗ ⟩ ⟨ X m , x ∗ ⟩ = 0 Then the normalized convex hulls W n = 1 21 n n 1 / 2 c o n v X 1 , ⋯ , X 1 converge with probability one in Hausdorff distance to the concentration ellipsoid of a limit Gaussian B-valued random element X . In addition, some related questions are discussed.