and computed the second moment of the difference of the expressions in (1) and (2). In this paper, by examining the excursions in Brownian motion and using a new formula for the distribution of their maxima, we obtain a direct identification of the limit in (1) without using (2). Let Tx=inf{t: W(t )=x} , Tx=inf{t: Y( t )=x} . For a > 0 let R~=0, R~= T/, + T/~ o Or,~, and for n> 2 let R~= R~,_ 1 + R~ o OR~_ 1 . Here {0~, t>0} is the usual collection of shift operators: W(s, Ot co) = W(s + t, co) and if S is a random variable, Os=O t on {S=t}. If S is a random variable, let d , (S )=sup{n: R~, 0 for ~ < s < fl; { Y(s, co), c~ < s < fl} is called an excursion if