Divergence criteria for random series related to the Azéma-Emery martingale process (Q1907443)

Consider the random series (1) \(\sum^\infty_{k=1} c^{2k} \xi_{k+1} \prod^k_{l=1} (1+ \xi_l)\), where \(c\) is a real nonzero constant and the sequence \(\{\xi_j: j= 1, 2,\dots\}\) is a sequence of independent and identically distributed random variables. The author studies necessary and sufficient conditions for the almost sure divergence of the above series. For example, if \(|c|\geq 1\) and the random variables \(\xi_k\) are positive, then the series (1) is divergent almost surely. To have more results, assume that \(|c|<1\), \(\xi_1> -1\) and that \(\ln (1+\xi_1)\in L^1\). Then the author shows that the series (1) converges almost surely, if \(E\ln c^2 (1+ \xi) <0\), and diverges almost surely, if \(E\ln c^2 (1+\xi_1)\geq 0\). The author also discusses the conservativity of the minimal dynamical semigroup associated with the Azema-Emery martingale in \(\mathbb{R}\) in connection with the convergence properties of (1).