Let $X_1,X_2,…$ be nonnegative random variables with partial sums $S_n=X_1+\cdots+X_n$. If $\lim_{n\to\infty} P(|\frac{S_n}{E(S_n)}−1|>\varepsilon)=0$ holds for any $\varepsilon>0$ (and thus the sequence $\frac{S_n}{E(S_n)}$ converges in probability to $X=1$), does this imply that $\lim_{n\to\infty}E|\frac{S_n}{E(S_n)}−1|^2=0$? If not, can you provide a counter example?
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Since $$p:=\mathbb{P}\left(\left|\frac{S_n}{\mathbb{E}(S_n)}-1\right|>\varepsilon\right) =\mathbb{P}\left(\left|\frac{S_n}{\mathbb{E}(S_n)}-1\right|^2>\varepsilon^2\right) $$ and $$ \mathbb{E}\left(\left|\frac{S_n}{\mathbb{E}(S_n)}-1\right|^2\right) \le (1-p)\varepsilon^2+p\mathbb{E}\left(\left|\frac{S_n}{\mathbb{E}(S_n)}-1\right|^2\big| \left|\frac{S_n}{\mathbb{E}(S_n)}-1\right|>\varepsilon\right). $$
Since by your assumption you have that $p\to 0$ if $n\to\infty$, it all depends on whether your sequence of partial sums is uniformly integrable. See also the answer to this question and the last property of this section on Wikipedia.
Bernhard
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A counterexample is given by choosing $X_i$ iid with $E[X_i]<+\infty$ and $E[X_i^2]=+\infty$. The law of large numbers give $S_n/E[S_n]\to 1$ in probability as required, but $$E[|S_n/E[S_n]-1|^2]=+\infty$$ for all $n$.
jlewk
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