The Saddle Point Method for the Integral of the Absolute Value of the Brownian Motion

is motivated by applications in order statistics: its distribution naturally appears in the consideration of the reliability of the procedure of determination of a theoretical distribution by drawing of a random sample. Namely, in the case when the sample size N is a Poissonian random variable, see Kac (1949a). The distribution function of the functional (1) was determined explicitly by Takács (1993) in the form of various series. The logarithmic tail asymptotics of this distribution follows from independent results by Borell (1975) and Kallianpur and Oodaira (1978). The purpose of the present paper is to outline how the saddle point method for integrals in the complex plane can be applied to determine the exact tail asymptotics of this distribution. These results where obtained in Tolmatz (1992), but remained unpublished. In the recent years the interest to distributions generated by various Wiener functionals is steadily growing; in particular we can mention applications to running times of certain algorithms and the applications in mathematical finance. Recently the asymptotics in question was computed by Fatalov (2003) by a completely different method. The corresponding results for the Brownian bridge in the L1 norm can be found in Cifarelli (1975), Shepp (1982), Johnson and Killeen (1983) and Tolmatz (2000). The method applied in Tolmatz (2000) is similar to the one used in Tolmatz (1992), but the case of Brownian motion is technically more complicated than that of the Brownian bridge, that is Tolmatz (2000) is a simplified version of Tolmatz (1992). On results regarding the Brownian bridge and Brownian motion in the L2 norm see Anderson and Darling (1952), Kiefer (1959), Tolmatz (2002), (2003), Cameron and Martin (1944).