When a sequence of random variables $(X_k)$ have different probability distributions, CLT is not necessarily stronger than LLN. The image above is captured from Feller Vol.1, P.255. This shows one example that CLT holds but LLN does not. The author says that the $\bf{sufficient}$ condition for LLN is $s_n/n \to 0$, where $s_n^2 = \sigma_1^2 + \cdots + \sigma_n^2$ ($Var(X_k) = \sigma_k^2$). Also, the Lindeberg theorem indicates that when $s_n \to \infty$ and $\frac{1}{s_n^2}\sum_{k=1}^n E(U_k^2) \to 1$, CLT can be applied. Here, $U_k$ is a truncated random variable, which is equal to $X_k$ if $|X_k| \le s_n \epsilon$ for any $\epsilon >0$, otherwise $U_k =0$.
I understand the example given here other than the statement which I highlight with purple line. I am confused whenever the author says "the order of magnitude …". What does it mean exactly in this context? In addition, I can't see how the author conclude that the law of large numbers cannot apply for $\lambda \ge 1/2$.
I would greatly appreciate if you give some help.
Best Answer
About the order of magnitude: it just means that the quotient $$ S_n/n^{\lambda+1/2} $$ is asymptotically (as $n\to\infty$) between two positive constants in probability. Therefore LLN cannot hold if $\lambda>1/2$ because in that case you would have $n^{\lambda+1/2}/n=n^{\epsilon}\to \infty$ as $n\to \infty$, which prevents $S_n/n$ to remain finite, close to the mean.