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On pages 46 and 51 of the book Statistical Inference based on divergence measures By Leandro Pardo Llorente there is a derivation for the Hellinger distance between two multivariate Gaussian distributions (the sample pages can be viewed on google books). If vectors of length 1 are used with this form and both distributions assumed to have zero means, then the exponential portion of the hellinger distance goes to unity. However this does not give the same result as the Hellinger distance between two univariate Gaussians under the same assumptions. Shouldn't these two forms be consistent under these assumptions?

Univariate form

Multivariate form

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They look the same, setting $\Sigma_{i}=\sigma_{i}^2$ for $i=1,2$... –  mjqxxxx Nov 24 '12 at 16:54
Whoops - yes your correct. Thanks. –  Mark Nov 24 '12 at 17:18

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