Convolution of a symmetric log-concave distribution and a symmetric bimodal distribution can have any number of modes

TL;DR

This paper demonstrates that convolving a symmetric log-concave distribution with a symmetric bimodal distribution can produce any number of modes, challenging assumptions about mode limitations in such convolutions.

Contribution

It establishes that the convolution of these specific symmetric distributions can have any positive number of modes, extending understanding of distribution mode behavior.

Findings

Convolution can produce any positive number of modes.

Results apply to both discrete and smooth distributions.

Challenges previous assumptions about mode limitations.

Abstract

In this note, we show that the convolution of a discrete symmetric log-concave distribution and a discrete symmetric bimodal distribution can have any strictly positive number of modes. A similar result is proved for smooth distributions.