2020/05/12 by Youri Davydov, Davydov, Youri, Vygantas Paulauskas +1
Decision Sciences · Economics, Econometrics and Finance · #FOS: Mathematics #Financial Risk and Volatility Modeling #Primary 60G15 #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #secondary 60F15
paper · pdf · doi:10.48550/arxiv.2005.05935
openalex publication_date 2020/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A "law of large numbers" for consecutive convex hulls for weakly dependent Gaussian sequences \Xn\, having the same marginal distribution, is extended to the case when the sequence \Xn\ has a weak limit. Let \mathbbB be a separable Banach space with a conjugate space \mathbbB^∗. Let \Xn\ be a centered \mathbbB-valued Gaussian sequence satisfying two conditions: 1) Xn ⇒ X and 2) For every x^* ∈ \mathbbB^∗ lim_ n,m, |n-m|→ ∞E⟨ Xn, x^*⟩ ⟨ Xm, x^*⟩ = 0. Then with probability 1 the normalized convex hulls Wn = \frac1(2ln n)1/2 \rm conv \ X1,…,Xn \ converge in Hausdorff distance to the concentration ellipsoid of a limit Gaussian \mathbbB-valued random element X. In addition, some related questions are discussed.