  An infinite bit sequence is called recursively random if no computable
strategy betting along the sequence has unbounded capital. It is well-known
that the property of recursive randomness is closed under computable
permutations. We investigate analogous statements for randomness notions
defined by betting strategies that are computable within resource bounds.
Suppose that S is a polynomial time computable permutation of the set of
strings over the unary alphabet (identified with N). If the inverse of S is not
polynomially bounded, it is easy to build a polynomial time random bit sequence
Z such that Z o S is not polynomial time random. So one should only consider
permutations S satisfying the extra condition that the inverse is polynomially
bounded. Now the closure depends on additional assumptions in complexity
theory.
  Our first result shows that if BPP contains a superpolynomial deterministic
time class then polynomial time randomness is not preserved by some permutation
S such that in fact both S and its inverse are in P. Our second result shows
that polynomial space randomness is preserved by polynomial time permutations
with polynomially boun\-ded inverse, so if P=PSPACE then polynomial time
randomness is preserved.
