Convergence in Total Variation to a Mixture of Gaussian Laws

Luca Pratelli and Pietro Rigo
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Luca Pratelli: Accademia Navale, Viale Italia 72, 57100 Livorno, Italy
Pietro Rigo: Dipartimento di Matematica “F. Casorati”, Universita’ di Pavia, via Ferrata 1, 27100 Pavia, Italy

Mathematics, 2018, vol. 6, issue 6, 1-14

Abstract: It is not unusual that X n ? d i s t V Z where X n , V , Z are real random variables, V is independent of Z and Z ∼ N ( 0 , 1 ) . An intriguing feature is that P V Z ∈ A = E N ( 0 , V 2 ) ( A ) for each Borel set A ⊂ R , namely, the probability distribution of the limit V Z is a mixture of centered Gaussian laws with (random) variance V 2 . In this paper, conditions for d T V ( X n , V Z ) → 0 are given, where d T V ( X n , V Z ) is the total variation distance between the probability distributions of X n and V Z . To estimate the rate of convergence, a few upper bounds for d T V ( X n , V Z ) are given as well. Special attention is paid to the following two cases: (i) X n is a linear combination of the squares of Gaussian random variables; and (ii) X n is related to the weighted quadratic variations of two independent Brownian motions.

Keywords: mixture of Gaussian laws; rate of convergence; total variation distance; Wasserstein distance; weighted quadratic variation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2018
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