Maximal inequalities for Bessel processes

It is proved that the uniform law of large numbers (over a random parameter set) for the alpha-dimensional (alpha greater than or equal to 1) Bessel process Z=(Z(t))(t greater than or equal to 0) started at 0 is valid: GRAPHICS for all stopping times T for Z. The rate obtained ton the right-hand side) is shown to be the best possible. The following inequality is gained as a consequence: GRAPHICS for all stopping times T for Z, where the constant G(alpha) satisfies GRAPHICS as alpha --> infinity. This answers a question raised in [4]. The method of proof relies upon representing the Bessel process as a time changed geometric Brownian motion. The main emphasis of the paper is on the method of proof and on the simplicity of solution.