An Optimal Selection Problem Associated with the Poisson Process

Cowan and Zabczyk (1978) introduced a continuous-time generalisation of the secretary problem, where offers arrive at epochs of a homogeneous Poisson process. We expand their work to encompass the last-success problem under the Karamata-Stirling success profile. In this setting, the $k$th trial is a success with probability $p_k=\theta/(\theta+k-1)$, where $\theta > 0$. In the best-choice setting ($\theta=1$), the myopic strategy is optimal, and the proof hinges on verifying the monotonicity of certain critical roots. We extend this crucial result to the last-success case by exploiting a connection to the sign of the derivative in the first parameter of a quotient of Kummer's hypergeometric functions. Additionally, we establish an Edmundson-Madansky inequality applicable to Poisson random variables. This result enables us to adopt a probabilistic approach to derive bounds and asymptotics of the critical roots. This strengthens and improves the findings of Ciesielski and Zabczyk (1979).