Stochastic models for single neuron firing trains: a survey.

Abstract

Theorem. If F is a df on [0,1], that is strictly increasing, and such that F(x) > x if O < x < ?, and F(x) < x if < x < 1, then F is not n-divisible for any n. There are similar, less simple results for df's with unbounded support. As a special case it follows that the uniform distribution is not n-divisible for any n though it is decomposable in many ways (see e.g., [2]). Kudina [1] uses a similar technique to prove the indecomposability of the arcsine df. It is well known that id df's with bounded support have zero variance, i.e., are degenerate. Similarly a df on [0,a] with variance exceeding a/(4n) is not ndivisible. This argument is extended to prove the following theorem (cf. [4], also for references).