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.