Let $X$ be a random variable dependend on some parameter $t$. with finite expectation and variance for all $t$. How can I compute $E[X(t) + o(t)]$ and $V[X(t) + o(t)]$ for $\rightarrow 0$?
From another question I asked here on MathSE, I could derive that its apparently true that $E[X(t) + o(t)] = E[X(t)] + o(t)$. But for the variance I am now sure whether my derivation is correct: $$V[X(t) + o(t)] = E[(X(t) + o(t))^2] - E[X(t) + o(t)]^2$$ so: $E[X(t) + o(t)]^2 = E[X(t)]^2 + E[X(t)]o(t) + o(t)^2$ but $E[X(t)]$ is constant and $\max\{o(t), o(t^2)\} = o(t)$ (as $t\rightarrow 0$), I should get $E[X(t) + o(t)]^2 = E[X(t)]^2 + o(t)$. For the other term I do the same computations to obtain: $E[(X(t) + o(t))^2] = E[X(t)^2] + o(t)$, so $$V[X(t) + o(t)] = E[X(t)^2] - E[X(t)]^2 + o(t) = V[X(t)] + o(t).$$ Is this correct?