If an update step described by a transition kernel 
is followed by
another step with transition kernel 
, then the combined step is
described by the transition kernel 
defined by
When the kernels are thought of as operators on a function space
then this multiplication of kernels corresponds to composition of operators
.
The notion of functional composition also describes the computer code. The composition of updates is implemented by following the code for the first update by the code for the second. If the code bits are thought of as implementing mathematical operations, then this also is a functional composition.
It is obvious from the definition that if 
is stationary for 
,
i = 1, 
, d. Then is also stationary for the composition
.