Some stochastic processes related to random density functions

Abstract

If {φ _n } is an orthonormal system and {a _n} is a sequence of random variables such that ∑ _n (a _n )²=1 a.s. thenf(t)=|∑ _n a _n φ _n (t)|² produces a randomly selcted density function. We study the properties off under the assumptions that |a _n| is decreasing to zero at a geometric rate and {φ _n } is one of the following four function systems: trigonometric Jacobi, Hermite, or Laguerre. It is shown that, with probability one,f is an analytic function,f has at most a finite number of zeros in any finite interval, and the tail off goes to zero rapidly.