$ \frac{(1+o_p(1) )A_n}{(1+o(1) )E[A_n]}=\frac{A_n}{E[A_n]}+ o_p(\frac{A_n}{E[A_n]}) ??$

Solution 1:

The term $ \frac{(1+o_p(1) )A_n}{(1+o(1) )E[A_n]}$ refers to something that can be expressed as $$ \frac{(1+Y_n )A_n}{(1+\delta_n )E[A_n]}, $$ where $(Y_n)_n$ is a sequence of random variables that converges in probability to $0$ and $(\delta_n)_n$ is a sequence of real numbers converging to 0$.

One has $$ \frac{(1+Y_n )A_n}{(1+\delta_n )E[A_n]}=\frac{A_n}{E[A_n] }+\frac{A_n}{E[A_n] }\underbrace{\left(\frac{1+Y_n }{1+\delta_n }-1\right)}_{o_p(1)}. $$ The term $o_p\left(\frac{A_n}{E[A_n]}\right)$ can therefore be expressed as $$ \frac{A_n}{E[A_n] }+\frac{A_n}{E[A_n] } \frac{Y_n-\delta_n }{1+\delta_n } . $$