On the Moments of Absolute Difference Between Random Processes With Application to Sensor Network

A closed analytical formula is given for expected absolute difference to the power
$a$
between two identical general random processes, when
$a$
is an even positive number. As an application to sensor network, we analyze the following scenario: Black colored sensors
$X_1,X_2,\dots,X_n$
and white colored sensors
$Y_1,Y_2,\dots,Y_n$
are initially randomly placed according to general random process. We minimize the energy consumption of the transportation cost to the power
$b>0$
of the maximal random bicolored matching.