# Probability Question (Expectation Value Limit)

Show that for any $\phi \in C_0(\mathbb{R^n})$ (a continuous function on $\mathbb{R^n}$ which tends to 0 at $\infty$), if we set

$\phi_\epsilon(x) = \mathbb{E}\phi(x + \epsilon Z)$

for any random variable $Z$ on $\mathbb{R^n}$, then $\phi_\epsilon(x)$ ? $\phi(x)$ as $\epsilon$ ? 0.

• Should we interpret x as being real valued and deterministic?

Answers can be viewed only if
1. The questioner was satisfied and accepted the answer, or
2. The answer was disputed, but the judge evaluated it as 100% correct.