Total variation convergence preserves conditional independence

Steffen Lauritzen

doi:10.1016/j.spl.2024.110200

Total variation convergence preserves conditional independence

Steffen Lauritzen

Department of Mathematical Sciences

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Abstract

This note establishes that if a sequence P_n,n=1,… of probability measures converges in total variation to the limiting probability measure P, and σ-algebras A and B are conditionally independent given H with respect to P_n for all n, then they are also conditionally independent with respect to the limiting measure P. As a corollary, this also extends to pointwise convergence of densities to a density.

Original language	English
Article number	110200
Journal	Statistics and Probability Letters
Volume	214
ISSN	0167-7152
DOIs	https://doi.org/10.1016/j.spl.2024.110200
Publication status	Published - 2024

Bibliographical note

Publisher Copyright:
© 2024 The Author(s)

Keywords

Markov properties
Scheffé’s theorem

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10.1016/j.spl.2024.110200Licence: CC BY

FulltextFinal published version, 45.6 KBLicence: CC BY

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abstract = "This note establishes that if a sequence Pn,n=1,… of probability measures converges in total variation to the limiting probability measure P, and σ-algebras A and B are conditionally independent given H with respect to Pn for all n, then they are also conditionally independent with respect to the limiting measure P. As a corollary, this also extends to pointwise convergence of densities to a density.",

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T1 - Total variation convergence preserves conditional independence

AU - Lauritzen, Steffen

PY - 2024

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N2 - This note establishes that if a sequence Pn,n=1,… of probability measures converges in total variation to the limiting probability measure P, and σ-algebras A and B are conditionally independent given H with respect to Pn for all n, then they are also conditionally independent with respect to the limiting measure P. As a corollary, this also extends to pointwise convergence of densities to a density.

AB - This note establishes that if a sequence Pn,n=1,… of probability measures converges in total variation to the limiting probability measure P, and σ-algebras A and B are conditionally independent given H with respect to Pn for all n, then they are also conditionally independent with respect to the limiting measure P. As a corollary, this also extends to pointwise convergence of densities to a density.

KW - Markov properties

KW - Scheffé’s theorem

UR - https://www.scopus.com/pages/publications/85197129167

U2 - 10.1016/j.spl.2024.110200

DO - 10.1016/j.spl.2024.110200

M3 - Journal article

AN - SCOPUS:85197129167

SN - 0167-7152

VL - 214

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

M1 - 110200

ER -

Total variation convergence preserves conditional independence

Abstract

Bibliographical note

Keywords

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