Does the Strong Law of Large Numbers Always Apply?

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SUMMARY

The Strong Law of Large Numbers (SLLN) applies to independent and identically distributed (iid) random variables, provided they have a finite mean and their second moments are bounded. However, the Kolmogorov Criterion offers a broader application, allowing for non-identical distributions as long as the series of variances converges. In cases where distributions are identical, only the mean is necessary, even if the variance is infinite. Understanding these conditions is crucial for correctly applying the SLLN in statistical analysis.

PREREQUISITES
  • Understanding of independent and identically distributed (iid) random variables
  • Knowledge of the Strong Law of Large Numbers (SLLN)
  • Familiarity with the Kolmogorov Criterion
  • Basic concepts of variance and convergence in series
NEXT STEPS
  • Research the implications of the Kolmogorov Criterion in probability theory
  • Study the conditions under which the Strong Law of Large Numbers fails
  • Explore examples of iid random variables and their applications
  • Learn about convergence of series and its relevance to statistical distributions
USEFUL FOR

Statisticians, data scientists, and mathematicians seeking to deepen their understanding of probability theory and the application of the Strong Law of Large Numbers in various statistical contexts.

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For the Strong Law of Large Numbers, as far as I know it applies when, let say, the random variables {Xn: n=1,2,...} are iid, hence uncorrelated, their second moments have a common bound and they have a finite mean mu.
What else I must consider? Is there anyother consideretion or case when the SLLN does not apply?
 
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There are weaker conditions for the strong law to hold.

For example, the Kolmogoroff Criterion does not require the distributions be identical.
It needs sum (Vk/k2) be a convergent series (V=variance).

Alternatively if the distributions are identical, all you need is the mean - the variance may be infinite.
 

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