  In this article, we introduce Skellam process of order k and its running
average. We also discuss the time-changed Skellam process of order k. In
particular we discuss space-fractional Skellam process and tempered
space-fractional Skellam process via time changes in Poisson process by
independent stable subordinator and tempered stable subordinator, respectively.
We derive the marginal probabilities, Levy measures, governing
difference-differential equations of the introduced processes. Our results
generalize Skellam process and running average of Poisson process in several
directions.
