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Convergence in Total Variation to a Mixture of Gaussian Laws

Author

Listed:
  • 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)

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.

Suggested Citation

  • Luca Pratelli & Pietro Rigo, 2018. "Convergence in Total Variation to a Mixture of Gaussian Laws," Mathematics, MDPI, vol. 6(6), pages 1-14, June.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:6:p:99-:d:151707
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    References listed on IDEAS

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    1. Nourdin, Ivan & Poly, Guillaume, 2013. "Convergence in total variation on Wiener chaos," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 651-674.
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