Dynamics of Bayes estimates for the rate of Poisson processes with gamma priors and convex loss (Q797945)

Let k(t): \(t>0\), be a continuous Poisson process with unknown rate \(\lambda\) and let \(d_{\gamma}(k(t),t)\) be the Bayes estimate of the rate \(\lambda\). The author studies the dynamic behavior of this Bayes estimate in the setting of the loss function of type \(|\lambda - d|^{\gamma}\), \(1<\gamma <\infty\). With a series of propositions and lemmas, the author finally proves a theorem, which states, that \(d_{\gamma}(k(t),t)\) is a sub-martingale for \(1<\gamma <2\), a martingale for \(\gamma =2\) and a super-martingale for \(\gamma >2\).