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An example of a non-log-concave distribution where the difference has a log-concave density

Min Wang

Abstract

By the Prékopa-Leindler inequality, the difference $X-X'$ has a log-concave density provided that $X$ has a log-concave density and $X, X'$ are independent and identically distributed. We prove that the opposite direction does not always hold true by giving an explicit example.

An example of a non-log-concave distribution where the difference has a log-concave density

Abstract

By the Prékopa-Leindler inequality, the difference has a log-concave density provided that has a log-concave density and are independent and identically distributed. We prove that the opposite direction does not always hold true by giving an explicit example.
Paper Structure (1 theorem, 3 equations)

This paper contains 1 theorem, 3 equations.

Key Result

Theorem 1

Let $X_1, X_2, X_3, X_4$ be independent standard normal distributions. Then $X_1 X_2 - X_3 X_4$ has a log-concave density, while $X_1 X_2$ has a non-log-concave density.

Theorems & Definitions (1)

  • Theorem 1