New representations and bounds for the generalized marcum Q-function via a geometric approach, and an application

The generalized Marcum Q-function of order m, Q_m(a, b), is interpreted geometrically as the probability of a 2m-dimensional, real, Gaussian random vector Z_2m, whose mean vector has a Frobenius norm of a, lying outside of a hyperball B_O,b ^2m of 2m dimensions, with radius b, and centered at the origin O. Based on this new geometric view, some new representations and closed-form bounds are derived for Q_m(a, b). For the case that m is an odd multiple of 0.5, a new closed-form representation is derived, which involves only simple exponential and ERFC functions. For the case that m is an integer, a pair of new, finite-integral representations for Q_m(a, b) is derived. Some generic exponential bounds and ERFC bounds are also derived by computing the probability of Z_2m lying outside of various bounding geometrical shapes whose surfaces tightly enclose, or are tightly enclosed by the surface of B_O,b ^2m. These bounding shapes consist of an arbitrarily large number of parts. As their closeness of fit with B_O,b ^2m improves, our generic bounds approach the exact value of Q_m(a, b). The function Q_m(a, b) is proved to be an increasing function of its order when 2m is a positive integer. Thus, Q_m+0.5(a, b) and Q_m-0.5(a, b) can be used as tight upper and lower bounds, respectively, on Q_m(a,b). Their average is a good approximation to Q_m(a, b). An application of our new representations and bounds is also given.