The level crossings of random sums

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.