For any $\mu>1/2$ let $Y_\mu \sim \mathrm{Poisson}(\mu)$ and let $c<\mu$.
Define $$ X_{\mu} = \frac{Y_\mu+c-\mu}{\mu}, $$ and then one can realise (pretty sure I have established this) that $$ \log(1+X_\mu) = X_\mu -\frac{1}{2}X_\mu^2 + \mathcal{O}_p(X_\mu^3), \quad \text{as } \mu\to\infty $$ which by definition of Big O' in probability means that $\forall \epsilon >0$ there exists $M>0$ and $\mu_0>0$ such that $$ P \left( \left| \frac{\log(1+X_\mu) -X_\mu + \frac{1}{2}X^2_\mu }{X^3_\mu} \right|> M \right) < \epsilon \quad \text{for all } \mu>\mu_0. $$
Now I want to establish some kind of normal Big O' statement of the expectation of this expression. More specifically I want something like
$$ E\log(1+X_\mu) = EX_\mu -\frac{1}{2}EX_\mu^2 + \mathcal{O}(EX_\mu^3), \quad \text{as } \mu\to \infty $$ because this would allow me to derive what I want to show, which is $$ E\log(1+X_\mu) = EX_\mu -\frac{1}{2}EX_\mu^2 + \mathcal{O}\left( \frac{1}{\mu^2} \right), \quad \text{as } \mu\to \infty, $$ since $EX_\mu^3 =\frac{c^3+3c\mu+\mu}{\mu^3} = \mathcal{O}\left( \frac{1}{\mu^2} \right)$ as $\mu\to \infty$.
Problem: How do I derive the $\mathcal{O}$ statement with the expectations from the $\mathcal{O}_p$ statement?