Necessary and Sufficient Condition for Identifying Independent Random Processes

A necessary and sufficient condition for an identifying independent random process is that the finite-dimensional probability density function of the process can be expressed with a product of some one-dimensional functions. The only difference between the one-dimensional functions and the corresponding marginal density function is a constant, and the boundary constants are irrelevant with variables and parameters. Those are two important characters of density functions of independent random processes. Using those characters to avoid solving the marginal density function makes the identification of an independence of random process simple.