Abstract
Brownian areas are considered in this paper: the Brownian excursion area, the Brownian bridge area, the Brownian motion area, the Brownian meander area, the Brownian double meander area, the positive part of Brownian bridge area, the positive part of Brownian motion area. We are interested in the asymptotics of the right tail of their density function. Inverting a double Laplace transform, we can derive, in a mechanical way, all terms of an asymptotic expansion. We illustrate our technique with the computation of the first four terms. We also obtain asymptotics for the right tail of the distribution function and for the moments. Our main tool is the two-dimensional saddle point method.
Highlights
Let Bex(t), t ∈ [0, 1], be a Brownian excursion, and let Bex :=Bex(t) dt be its area
It is known that Bex has a density function fex, which was given explicitly by Takacs [22] as a convergent series involving the zeros aj of the Airy function and the confluent hypergeometric function U :
We define here the Brownian double meander by Bdm(t) := B(t)−min0≤u≤1 B(u); this is a non-negative continuous stochastic process on [0, 1] that a.s. is 0 at a unique point τ ∈ [0, 1], and it can be regarded as two Brownian meanders on the intervals [0, τ ] and [τ, 1] joined back to back, see Majumdar and Comtet [17] and Janson [10]; the other processes considered here are well-known, see for example Revuz and Yor [20]. We find it illuminating to study all seven Brownian areas together, and we will formulate our proof in a general form that applies to all seven areas
Summary
An alternative way to obtain (1.4) is by large deviation theory, which gives (1.4) and explains the constant 6 as the result of an optimization problem, see Fill and Janson [7] This method applies to the other Brownian areas in this paper too, and explains the different constants in the exponents below, but, again, it seems difficult to obtain more precise results by this approach. There are differences in factors of x between Bbr and Bex, and between Bbm and Bme, where the process conditioned to be positive has somewhat higher probabilities of large areas These differences are in the expected direction, but we see no intuitive reason for the powers in the theorems. For example, C1(M ) to denote dependency on a parameter (but not on anything else)
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