A note on quenched moderate deviations for Sinai’s random walk in random environment

Francis Comets; Serguei Popov

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In this paper, we consider a random entire function $f (s, ω)$ defined by a random Dirichlet series $\sum_{n = 1}^{\infty} X_{n} (ω) e^{- λ_{n} s}$ where $X_{n}$ are independent and complex valued variables, $0 \leq λ_{n} ↗ + \infty$ . We prove that under natural conditions, for some random entire functions of order $(R)$ zero $f (s, ω)$ almost surely every horizontal line is a Julia line without an exceptional value. The result improve a theorem of J. R. Yu: Julia lines of random Dirichlet series. Bull. Sci. Math. 128 (2004), 341–353, by relaxing condition on the distribution of...