About Closedness by Convolution of the Tsallis Maximizers

In this paper, we study the stability under convolution of the maximizing distributions of the Tsallis entropy under energy constraint (called hereafter Tsallis distributions). These distributions are shown to obey three important properties: a stochastic representation property, an orthogonal invariance property and a duality property. As a consequence of these properties, the behaviour of Tsallis distributions under convolution is characterized. At last, a special random convolution, called Kingman convolution, is shown to ensure the stability of Tsallis distributions.