# References for Kolmogorov's strong law of a large numbers

On the Wikipedia law of large numbers site, they mention "Kolmogorov's strong law of large numbers", which works even if the random variables are not identically distributed.

Where can I find this theorem shown and proven? I know that a reference is provided on the Wikipedia site, but that book is out of availability. Are there any other references out there?

(Interestingly, Allan Gut's book "Probability: A Graduate Course", has a theorem by the name of "Kolmogorov's strong law", but in his book, the random variables have to be identically distributed. Any ideas why this is?)

-

## 1 Answer

It does not work, in general, if the summands are not iid. Both "independent" and "identically distributed" can be weakened, but you can't dispense with either of them entirely and still get the result without giving up something else.

The result that they cite as "Kolmogorov's Strong Law" is not what I always refer to as Kolmogorov's Strong Law (I suspect that what I refer to as Kolmogorov's Strong Law is the same thing that Allan Gut does). The result given on Wikipedia requires a finite second moment and that $\sum \frac{\mbox{Var} X_k}{k^2} < \infty$, but in exchange you can drop the requirement of being identically distributed. The proof of this version I think is actually pretty easy, the sketch being: because the series converges we can apply the Khintchine-Kolmogorov convergence theorem so that $\sum \frac{(X_k - \mu_k)}{k}$ converges almost surely, and the result follows after an application of Kronecker's Lemma.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9389640092849731, "perplexity_flag": "head"}