The monotonicity of the threshold detection probability in a stochastic accumulation process

In this paper we prove for a number of distributions that the probability for the value of the sum of the first k (but not before) of i.i.d.r.v. to exceed a given value A is monotonically increasing in the range k < k^* (or k < k^* + 1 ) where k^* = max k such that kμ ≤A. We conjecture that this monotonicity property is preserved for a much larger family of distribution functions than those examined in the paper.