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arXiv · 2512.15128

On non-stationarity of the Poisson gamma state space models

Kaoru Irie·Tevfik Aktekin
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Abstract

The Poisson-gamma state space (PGSS) models have been utilized in the analysis of non-negative integer-valued time series to sequentially obtain closed form filtering and predictive densities. In this study, we show the underlying mechanics and non-stationary properties of multi-step ahead predictive distributions for the PGSS family of models. By exploiting the non-stationary structure of the PGSS model, we establish that the predictive mean remains constant while the predictive variance diverges with the forecast horizon, a property also found in Gaussian random walk models. We show that, in the long run, the predictive distribution converges to a zero-degenerated distribution, such that both point and interval forecasts eventually converge towards zero. In doing so, we comment on the effect of hyper-parameters and the discount factor on the long-run behavior of the forecasts.

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Kaoru Irie, Tevfik Aktekin. 2025-12-17. On non-stationarity of the Poisson gamma state space models. https://arxiv.org/abs/2512.15128

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