It is proved that the uniform law of large numbers (over a random para
meter 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 i
s valid: GRAPHICS for all stopping times T for Z. The rate obtained to
n the right-hand side) is shown to be the best possible. The following
inequality is gained as a consequence: GRAPHICS for all stopping time
s 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 geometri
c Brownian motion. The main emphasis of the paper is on the method of
proof and on the simplicity of solution.