Note on the moment generating function of the multivariate normal distribution

Hirose, Kenichi (2024) Note on the moment generating function of the multivariate normal distribution. Miskolc Mathematical Notes, 25 (1). pp. 287-300. ISSN 1787-2413

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Abstract

We present a streamlined proof of a formula for the derivatives of the moment generating function of the multivariate normal distribution. We formulate it in terms of the summation of the contractions by pairings, which encodes a combinatorial computation procedure. We give two applications. First, we provide a simple proof of Isserlis’ theorem and derive a formula for the moments of the multivariate normal distribution. Second, we calculate the moments of the product of a finite number of correlated normally and lognormally distributed random variables.

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