  Let $\{\eta_{j}\}_{j = 0}^{N}$ be a sequence of independent and identically
distributed complex normal random variables with mean zero and variances
$\{\sigma_{j}^{2}\}_{j = 0}^{N}$. Let $\{f_{j} (z)\}_{j = 0}^{N}$ be a sequence
of holomorphic functions that are real-valued on the real line. The purpose of
the present study is that of examining the number of times that the random sum
$\sum_{j = 0}^{N} \eta_{j} f_{j} (z)$ crosses the complex level $\boldsymbol{K}
= K_{1} + i K_{2}$, where $K_{1}$ and $K_{2}$ are constants independent of $z$.
More specifically, we establish an exact formula for the expected density
function for the complex zeros. We then reformulate the problem in terms of
successive observations of a Brownian motion. We further answer the basic
question about the expected number of complex zeros for coefficients of
nonvanishing mean values.
