Arbitrarily Tight Upper and Lower Bounds on the Gaussian Q-function and Related Functions

We present a new family of tight lower and upper bounds on the Gaussian Q-function Q(x). It is first shown that, for any x, the integrand phi(theta; x) of the Craig representation of Q(x) can be partitioned into a pair of complementary convex and concave segments. As a consequence of this property, integrals of phi(theta; x) over arbitrary intervals within its convex region can be lower-bounded by Jensen's inequality and upper-bounded by Cotes' quadrature rule, with the opposite occurring for the concave region phi(theta; x). The combination of these complementary bounds yield a complete family of both lower and upper bounds on Q(x), which are expressed in terms of elementary transcendental functions and can be made arbitrarily tight by finer segmentation. A by-product of the method is that various other functions, such as the squared Gaussian Q-function Q(2)(x), the 2D joint Gaussian Q-function Q (x, y, p), and the generalized Marcum Q-function Q(M)(x, y), can also be both upper and lower bounded with arbitrarily tightness, which to the best of our knowledge finds no precedence in the literature. Explicit examples of the latter applications are given.