Picard points of random Dirichlet series (Q1568131)

Let \(f(s,\omega)=\sum_{n=0}^\infty a_nZ_n(\omega)\exp(-s\lambda_n)\) be a random Dirichlet series such that: (1) \(0=\lambda_0<\lambda_n\nearrow+\infty\), \(\limsup_{n\to+\infty}\frac{\log n}{\lambda_n}=0\), (2) \((a_n)_{n=0}^\infty\subset\mathbb C\), \(\limsup_{n\to+\infty}\frac{\log|a_n|}{\lambda_n}=0\), \(\limsup_{n\to+\infty}\frac{\log|a_n|}{\log\lambda_n}=\infty\), (3) \((Z_n)_{n=0}^\infty\) is a sequence of non-degenerate independent equally-distributed random variables on a complete probability space, \(\limsup_{n\to+\infty}\frac{\log|Z_n|}{\lambda_n}\leq 0\) a.s. Then the series is a.s. convergent for \(\operatorname{Re}s>0\) and \[ \text{for }\varepsilon>0,\;t_0\in{\mathbb R},\;a\in{\mathbb C}: \limsup_{\sigma\to 0+} \sigma n(\sigma;t_0-\varepsilon, t_0+\varepsilon; f(s,\omega)=a)=\infty \text{ a.s.}, \] where \(n(\sigma;t_0-\varepsilon, t_0+\varepsilon; f(s,\omega)=a):=\# \{s:\operatorname {Re}s>\sigma,\;|\)Im\(s-t_0|<\varepsilon.f(s,\omega)=a\}\).