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 \emph{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 $\tilde{\mathcal{O}}(ε^{-3})$ sample complexity for TSIVR-PG to find an $ε$-stationary policy. By assuming the overparameterizaiton of policy and exploiting the hidden convexity of the problem, we further show that TSIVR-PG converges to global $ε$-optimal policy with $\tilde{\mathcal{O}}(ε^{-2})$ samples.