Convergence in Total Variation to a Mixture of Gaussian Laws
Accademia Navale, Viale Italia 72, 57100 Livorno, Italy
Dipartimento di Matematica “F. Casorati”, Universita’ di Pavia, via Ferrata 1, 27100 Pavia, Italy
Author to whom correspondence should be addressed.
Received: 29 April 2018 / Revised: 1 June 2018 / Accepted: 5 June 2018 / Published: 11 June 2018
It is not unusual that
are real random variables, V
is independent of Z
. An intriguing feature is that
for each Borel set
, namely, the probability distribution of the limit
is a mixture of centered Gaussian laws with (random) variance
. In this paper, conditions for
are given, where
is the total variation distance between the probability distributions of
. To estimate the rate of convergence, a few upper bounds for
are given as well. Special attention is paid to the following two cases: (i)
is a linear combination of the squares of Gaussian random variables; and (ii)
is related to the weighted quadratic variations of two independent Brownian motions.
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MDPI and ACS Style
Pratelli, L.; Rigo, P. Convergence in Total Variation to a Mixture of Gaussian Laws. Mathematics 2018, 6, 99.
Pratelli L, Rigo P. Convergence in Total Variation to a Mixture of Gaussian Laws. Mathematics. 2018; 6(6):99.
Pratelli, Luca; Rigo, Pietro. 2018. "Convergence in Total Variation to a Mixture of Gaussian Laws." Mathematics 6, no. 6: 99.
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