Hankel Determinants of Random Moment Sequences

Holger Dette () and Dominik Tomecki ()
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Holger Dette: Ruhr-Universität Bochum
Dominik Tomecki: Ruhr-Universität Bochum

Journal of Theoretical Probability, 2017, vol. 30, issue 4, 1539-1564

Abstract: Abstract For $$t \in [0,1]$$ t ∈ [ 0 , 1 ] let $$\underline{H}_{2\lfloor nt \rfloor } = (m_{i+j})_{i,j=0}^{\lfloor nt \rfloor }$$ H ̲ 2 ⌊ n t ⌋ = ( m i + j ) i , j = 0 ⌊ n t ⌋ denote the Hankel matrix of order $$2\lfloor nt \rfloor $$ 2 ⌊ n t ⌋ of a random vector $$(m_1,\ldots ,m_{2n})$$ ( m 1 , … , m 2 n ) on the moment space $$\mathcal {M}_{2n}(I)$$ M 2 n ( I ) of all moments (up to the order 2n) of probability measures on the interval $$I \subset \mathbb {R}$$ I ⊂ R . In this paper we study the asymptotic properties of the stochastic process $$\{ \log \det \underline{H}_{2\lfloor nt \rfloor } \}_{t\in [0,1]}$$ { log det H ̲ 2 ⌊ n t ⌋ } t ∈ [ 0 , 1 ] as $$n \rightarrow \infty $$ n → ∞ . In particular weak convergence and corresponding large deviation principles are derived after appropriate standardization.

Keywords: Hankel determinant; Random moment sequences; Weak convergence; Large deviation principle; Canonical moments; Arcsine distribution; 60F05; 60F10; 30E05; 15B52 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10959-016-0699-z

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