Strong law of large numbers for a scaled sequence of normally distributed random variables
Solution 1:
I guess the claim is wrong in general. However, if we assume that $g'$ is Lipschitz continuous, we're able to conclude $f(x)\xrightarrow{|x|\to\infty}0$ and $f'(x)\xrightarrow{|x|\to\infty}0$. If we now further assume that $$\int|f''|\:{\rm d}\lambda^1<\infty\tag6,$$ we obtain $$\int fg''\:{\rm d}\lambda^1\xleftarrow{r\to\infty}f'(r)-f'(-r)-\int_{-r}^rf'(x)g'(x)\:{\rm d}x\xrightarrow{r\to\infty}-I\tag7$$ by Lebesgue's dominated convergence theorem and partial integration. In particular, $$\int f''\:{\rm d}\lambda^1=0\tag8.$$