Date of Award


Document Type


Degree Name

Doctor of Philosophy (PhD)

Legacy Department

Mathematical Science


Lund, Robert B

Committee Member

Fralix , Brian

Committee Member

Gallagher , Colin

Committee Member

Kiessler , Peter


Most data collected over time has some degree of periodicity (i.e. seasonally varying
traits). Climate, stock prices, football season, energy consumption, wildlife sightings, and
holiday sales all have cyclical patterns. It should come as no surprise that models that
incorporate periodicity are paramount in the study of time series.
The following work devises time series models for counts (integer-values) that are periodic
and stationary. Foundational work is rst done in constructing a stationary periodic
discrete renewal process (SPDRP). The dynamics of the SPDRP are mathematically interesting
and have many modeling applications, expositions largely unexplored here. This work
develops a SPDRP as a generation mechanism to produce a stationary count time series
models with many desirable characteristics, including periodicity, negative autocovariances
and long-memory.
After development of the SPDRP univariate count models are generalized into multiple
dimensions. A multivariate renewal process has many interrelated stochastic processes.
The resulting multivariate model has all the desirable properties of its univariate kin, but
can also have negative autocovariances between marginal components of the series. To our
knowledge, this trait is seldom achieved in current multivariate count methods in tandem
with long-memory and periodicity