# Analytic Properties of Random Processes

Properties :: Continuity :: Differentiation :: Integration

## Continuity

Let
. Recall that a function
is
**
continuous
**
at
if
.
is continuous if it is continuous for every
.

Continuity w.p. 1 is usually quite strong, in fact stronger than is
typically needed for analysis.

We say that
is
**
continuous in probability
**
at
if
(i.p.) as
.
That is,

We say that is

**mean-square continuous**at if (m.s.) as . That is,

An important property: A second-order random process is mean-square continuous at if and only if is continuous at , that is

For a W.S.S. process, we have the following: A W.S.S. r.p.
is
mean-square continuous if and only if
is continuous at
. This follows since
, and