# A Note on Contiguity and Hellinger Distance

• J. Oosterhoff
• W. R. van Zwet
Open Access
Chapter
Part of the Selected Works in Probability and Statistics book series (SWPS)

## Abstract

For n = 1, 2, … let (X n1, A n1), …, (X nn , A nn ) be arbitrary measurable spaces. Let P ni and Q ni be probability measures defined on (X ni , A ni ), i = 1, …, n; n = 1, 2, …, and let $$P_n^{\left( n \right)} = \prod\limits_{i = 1}^n {{P_{ni}}}$$ and $$Q_n^{\left( n \right)} = \prod\limits_{i = 1}^n {{Q_{ni}}}$$ denote the product probability measures. For each i and n let X ni be the identity map from X ni onto X ni . Then P ni and Q ni represent the two possible distributions of the random element X ni as well as the probability measures of the underlying probability space. Obviously X n1, …, X nn are independent under both $$P_n^{\left( n \right)}$$ and $$Q_n^{\left( n \right)}$$ (n = 1, 2, …).

### Key Words & Phrases

asymptotic normality contiguity Hellinger diftance log likelihood ratio

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