  In this paper, we mainly focus on how to generalize some conclusions from
nonnegative irreducible tensors to nonnegative weakly irreducible tensors. To
do so, a basic and important lemma is proven using new tools. First, we give
the definition of stochastic tensors. Then we show that every nonnegative
weakly irreducible tensor with spectral radius being one is diagonally similar
to a unique weakly irreducible stochastic tensor.
  Based on it, we prove some important lemmas, which help us to generalize the
results related. Some counterexamples are provided to show that some
conclusions for nonnegative irreducible tensors do not hold for nonnegative
weakly irreducible tensors.
