  Lazarsfeld and Mustata propose general and systematic usage of Okounkov's
idea in order to study asymptotic behavior of linear series on an algebraic
variety. It is a very simple way, but it yields a lot of consequences, like
Fujita's approximation theorem. Yuan generalized this way to the arithmetic
situation, and he established the arithmetic Fujita's approximation theorem,
which was also proved by Chen independently. In this paper, we introduce
arithmetic linear series and give a general way to estimate them based on
Yuan's idea. As an application, we consider an arithmetic analogue of the
algebraic restricted volumes.
