I am trying to figure out the Stein's identity which asserts that for r.v $X$ having pdf $$p_\theta(x)=\exp\{ \theta T(x)-A(\theta)\}h(x)$$ where $ T$ is differentiable and $g>0$ is differentiable s.t. $$E(g(X)), E(|g'(X)|) < \infty,$$ then $$E\left(\left[\frac{h'(X)}{h(X)}+\theta T'(X)\right]g(X)\right)=-E(g'(X))$$ provided the support of $X$ is $(-\infty,\infty)$.
The method is integration by part but one needs to show that $$\lim_{x\rightarrow \infty} p_\theta(x)g(x) =0.$$
The condition $E(g(X)) < \infty$ alone seems not enough ( there could be some spikes of $p_\theta g$) but I do not know how to utilize the condition $E(g'(X))< \infty$.
I have also found if $p_\theta g$ is uniform continuous, then $p_\theta g$ would approach 0 but how to show the uniform continuity then?
or $p_\theta g$ may not approach 0?
Any help would be appreciated!