A Normal Limit Theorem for Power Sums of Independent Random Variables

In many areas of transmission engineering, logarithms of sums of powers are considered in the form 1 = 10 Iog10 [10- W10 + · · · + 10 W 1 ° ] , where X1, . . ., X,, are random variables. Specifically, if A'i, . . are power levels in dB such that X,- = 10 log10 (w,/w u ) j = 1,2, ,n, Ar,, where w0 , wx , · · · , w,, are powers (e.g., expressed in watts), then the power level in dB of the sum w = wl + · · * + w,, is given by the socalled "power sum," Pn = 10 log10 (w/w0) = 10 log10 [10 Y,/1 ° + · · · + 10- W10 ]. Quite often X^, · · · , X,, arc taken to be mutually independent random variables with specified distributions, and it is of interest to determine properties of their power sum P,,. A major difficulty encountered in working with power sums is that the distribution and moments of such a sum usually cannot be ex2081 2084 T H E BELL SYSTEM T E C H N I C A L J O U R N A L , NOVEMBER 19(57 pressed in simple closed form. This includes, for example, the important case when X, · · · , Xn are mutually independent and each has a truncated normal distribution. Even in the simpler case when Xlf · · · , Ar,, are mutually independent, identically distributed, and Xx is normal, the problem is intractable. The difficulty and importance of the general problem, in turn, has led to a number of methods for approximating the distribution of a power sum.1-2> 3*4> 0-0i 8> 9 In the present paper, the asymptotic distribution of a power sum is studied. The main result is a limit theorem which shows that under very general conditions on the components Xi, X2, · · · , the corresponding power sums P,, will be asymptotically normal as n --» oo.