The level crossings of random sums

Abstract

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 $\mathbf{K}=K_1+i{K}_2$, 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.

16 pages