论文标题
随机总和的水平交叉
The level crossings of random sums
论文作者
论文摘要
令$ \ {η_{j} \} _ {j = 0}^{n} $为独立且相同分布的复杂正常随机变量的序列，平均为零，方差$ \ \ \ {σ_{J}^{2}^{2}^{2} {2} \} _ {J = 0} _ {令$ \ {f_ {j}（z）\} _ {j = 0}^{n} $是一系列holomorphic函数，在真实行中实现了现实价值。本研究的目的是检查随机总和$ \ sum_ {j = 0}^{n}η_{ $ z $。更具体地说，我们为复杂零的预期密度函数建立了一个确切的公式。然后，我们从连续观察布朗动议的角度重新制定了问题。我们进一步回答了关于不变平均值系数的复杂零的预期数量的基本问题。
Let $\{η_{j}\}_{j = 0}^{N}$ be a sequence of independent and identically distributed complex normal random variables with mean zero and variances $\{σ_{j}^{2}\}_{j = 0}^{N}$. 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 $\sum_{j = 0}^{N} η_{j} f_{j} (z)$ 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.
