On modality of Lévy processes corresponding to mixtures of two exponential distributions (Q1908708)

Following \textit{K. Sato} [Theory Probab. Appl. 39, No. 2, 336-347 (1994) and Teor. Veroyatn. Primen. 39, No. 2, 403-414 (1994; Zbl 0834.60018)], the author looks for Lévy processes \(X(t)\) whose distributions have different modality for different values of \(t\). Taking \(X(1)\) to have a distribution with density \((1 - p)ae^{-ax} + pbe^{-bx}\), with \(0 < a < b\) on \((0,\infty)\), he proves the following result: \(X(n)\) is either unimodal or bimodal; it is unimodal if \(n \geq n(a,b,p)\). Here \(n(a,b,p)\) is a simple function of \(a\), \(b\) and \(p\). \(X(n)\) is not necessarily bimodal for \(n < n(a,b,p)\), but it will be bimodal for fixed \(n\) and large values of \(b\).