On a study of the exponential and Poisson characteristics of the Poisson process

Based on the exponential and Poisson characteristics of the Poisson process, in this work we present some characterizations of the Poisson process as a renewal process. More precisely, let γt be the residual life at time t of the renewal process A = {A(t), t ≥ 0}, under suitable condition, we prove that if Var(γt) = E²(γt), ∀t ≥ 0, then A is a Poisson process. Secondly, we show that if Var(A(t)) is proportional to E(A(t)), then A is a Poisson process also, and Var(A(t)) = E(A(t)).