The random seed is typically a short binary string drawn from the uniform distribution.
Many different classes of statistical tests have been considered in the literature, among them the class of all Boolean circuits of a given size. It is not known whether good pseudorandom generators for this class exist, but it is known that their existence is in a certain sense equivalent to (unproven) circuit lower bounds in computational complexity theory. Hence the construction of pseudorandom generators for the class of Boolean circuits of a given size rests on currently unproven hardness assumptions.
Generally, Pseudorandom generators are not suitable for Cryptography as the output of a Cryptographically secure pseudorandom number generator must guarantee the output is random.
In Cryptography discussions we typically make a Computational Hardness Assumption.