A note on seasonal Markov chains with gamma or gamma-like distributions

Abstract

Weighted sums defined on a Markov chain (MC) are important in applications (e.g. to reservoir storage theory). The rather intractable theory of such sums simplifies to some extent when the transition p.d.f. of the chain { X t } has a Laplace transform (LT) L ( X t +1 ; θ |Χ t =x ) of the ‘exponential' form H ( θ ) exp{ – G (θ) x }. An algorithm is derived for the computation of the LT of Σ a t ,Χ t for this class, and for a seasonal generalization of it. A special case of this desirable exponential type of transition LT for a continuous-state discrete-time MC is identified by comparison with the LT of the Bessel distribution. This is made the basis for a new derivation of a gamma-distributed MC proposed by Lampard (1968). A seasonal version of this process is developed, valid for any number of seasons. Reference is made to related chains with three-parameter gamma-like distributions (of the Kritskii–Menkel family) that may be generated from the above by a simple power transformation.