Asymptotics of Fisher Information Under Weak Perturbation

An asymptotic expression as $\varepsilon\to 0$ is derived for the Fisher information of a random variable $Y=X+Z_\varepsilon$, where $X$ and $Z_\varepsilon$ are mutually independent, under some regularity conditions on the probability density function of $X$ and the assumption that $\mathbf{E}Z^2_\varepsilon=\varepsilon^2$ and $\mathbf{E}|Z_\varepsilon/\varepsilon|^k\leq c<\infty$ for some $k>2$. Using this result an asymptotic generalization of De Bruijn's identity is obtained.