Final answer:
Mean-square convergence does not necessarily imply almost sure convergence.
While the Central Limit Theorem informs us about the distribution of sample means, it is not directly related to almost sure convergence.
Step-by-step explanation:
Mean-square convergence is a type of convergence in probability theory where the means of successive random variables get closer in a squared error sense.
While this type of convergence does relate to the behavior of the variances and averages of random variables, it does not guarantee that the outcomes will converge to a particular value with certainty, which is required for almost sure convergence.
Moreover, the Central Limit Theorem (CLT) is often discussed in this context. The CLT states that, under certain conditions, the distribution of sample means approaches a normal distribution as the sample size gets larger, regardless of the population's distribution.
This is often used to help in understanding the distribution of sample means and sums, but it doesn't directly provide information about the convergence of individual sample paths.