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It is stated on wikipedia that:

"Lindeberg's condition is sufficient, but not in general necessary (i.e. the inverse implication does not hold in general). However, if the sequence of independent random variables in question satisfies $$ \max_{k=1,\ldots,n} \frac{\sigma_k^2}{s_n^2} \to 0, \quad \text{ as } n \to \infty $$ then Lindeberg's condition is both sufficient and necessary, i.e. it holds if and only if the result of central limit theorem holds."

My question is regarding what it means for the condition to be sufficient but not necessary on its own. What does it mean for the inverse implication not to hold in general? Does it mean that a central limit not holding implies the Lindeberg condition being false?

Additionally, if the condition is both sufficient and necessary, why does the hold if and only if the central limit theorem holds?

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