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

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.

Figures (5)

• Figure 1: Case $n=6$. The $\bullet$ correspond to $\tilde{q_6}$, the $\times$ to $\tilde{p_6}$ and the $+$ to where they take the same value.
• Figure 2: The convolution product $\tilde{p_n}*\tilde{q_n}$ in the case $n=6$.
• Figure 3: Case $n=7$. The $\bullet$ correspond to $\tilde{q_7}$, the $\times$ to $\tilde{p_7}$ and the $+$ to where they take the same value.
• Figure 4: The graph of $h_N:x \mapsto \frac{1}{1+\exp(\frac{Nx}{x^2-1})}1_{\{-1< x < 1\}} + 1_{\{x\geq1\}}$ for $N=10$.
• Figure 5: The graph of $\tilde{g}_n^N$, which serves as a smooth approximation of $\Phi(\tilde{q}_n)$, for $n=5$ and $N=10$.

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Remark 2.1
• proof : Proof of Theorem \ref{['TheoremDiscret']}
• Remark 3.1
• Lemma 4.1
• Lemma 4.2
• proof : Proof of Lemma \ref{['TechnicalLemma2']}
• proof : Proof of Theorem \ref{['TheoremContinu']}