Policy gradient (PG) gives rise to a rich class of reinforcement learning (RL) methods. Recently, there has been an emerging trend to accelerate the existing PG methods such as REINFORCE by the variance reduction techniques. However, all existing variance-reduced PG methods heavily rely on an uncheckable importance weight assumption made for every single iteration of the algorithms. In this paper, a simple gradient truncation mechanism is proposed to address this issue. Moreover, we design a Truncated Stochastic Incremental Variance-Reduced Policy Gradient (TSIVR-PG) method, which is able to maximize not only a cumulative sum of rewards but also a general utility function over a policy's long-term visiting distribution. We show an Õ(ǫ−3) sample complexity for TSIVR-PG to find an ǫ-stationary policy. By assuming the overparameterization of policy and exploiting the hidden convexity of the problem, we further show that TSIVR-PG converges to global ǫ-optimal policy with Õ(ǫ−2) samples.