  It is well known that under some conditions the almost sure survival
probability of a multitype branching processes in random environment is
positive if the Lyapunov exponent corresponding to the expectation matrices is
positive, and zero if the Lyapunov exponent is negative. The goal of this note
is to establish similar results when certain positivity conditions on the
expectation matrices are not met. One application of such a result is to
classify the positivity of Lebesgue measure of certain overlapping random
self-similar sets in the line.
