Lower bound for the diameter of planar Brownian motion

Let B(t) be a standard planar Brownian motion and r(theta) be the diameter of the projection of B(Left perpendicular 0, 1 Right perpendicular) on the line generated by the unit vector e(theta) = (cos theta; sin theta), where 0 <= theta <= pi. In this short note, we find the common cumulative distribution function F of the random variables r(theta). Namely, we prove that F(x) = 8 Sigma(infinity)(n=1)(1/x(2) + 1/(2n - 1)(2)pi(2)) exp (-(2n - 1)(2)pi(2)/2x(2)), for every x > 0. As immediate consequence, lower bound for the expected diameter of the set B([0, 1]), better than known, is obtained. Namely, it is known that Ed >= 1.601, where d is the diameter of the set B([0, 1]). In this note we show Ed >= 1:856.