On Two-dimensional Brownian Motion In The Possession Of The Tapered Region

It is known that the occupation time in the positive axis for a standard onedimensional Brownian motion follows the famous L(?)vy's are-sine law,and that in general intervals can be obtained explicitly or indirectly,while there are much less corresponding results in higher dimensions.We consider the occupation time in a cone in the positive quadrant for a standard two-dimensional Brownian motion,which is still an open problem until now,but we can attempt to study it from other aspects.Bingham and Doney(1988) considered the occupation time in the positive quadrant,and obtained expressions of the first,second and third moments,then negated possible conjectures about the distributions of the occupation time.We generalize their results following the original question,and obtain expressions of the first and second moments.In addition, we indicate this sort of occupation time never has the generalized are-sine distribution.