In this letter, we derive new results for the statistics of the complex random variablez=DeltaSigman=1Nxn2=z1+jzQ=rejphiwhere {xn} is a set of mutually independent complex-valued Gaussian random variables with arbitrary means and equal variances. Each random variablexnis assumed to have independent real and imaginary components with equal variance for alln. Expressions are derived for the joint probability density function (pdf) of (z1,zQ), for the joint pdf of (r,phi) and also for the marginal pdf of the modulusr. An useful Fourier series expansion for the pdf of the phase phi is also derived. As an application of the results, a theoretical performance analysis of the well-known nondata-aided Viterbi and Viterbi feedforward carrier phase estimator operating with BPSK signals is presented. In particular, the expressions for the exact pdf, variance, and equivocation probability of the carrier phase estimates are derived.