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.