Geodesic
On conditions for SLLN for martingales with identically distributed increments
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 4 (2007), pp. 547-552 
A. I. Sakhanenko. On conditions for SLLN for martingales with identically distributed increments. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 4 (2007), pp. 547-552. http://geodesic.mathdoc.fr/item/SEMR_2007_4_a29/
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     author = {A. I. Sakhanenko},
     title = {On conditions for {SLLN} for martingales with identically distributed increments},
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     url = {http://geodesic.mathdoc.fr/item/SEMR_2007_4_a29/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

For any random variable $X$ with $\mathbf E\big[|X|\log(1+|X|)\big]=\infty$ and $\mathbf{E}X=0$ we construct a sequence $\{X_n:n\ge1\}$ of martingale differences which are identically distributed with $X$ and such that the strong law of large numbers does not hold.

[1] P. Hall, C. C. Heyde, Martingale Limit Theory and Its Application, Academic Press, New York, 1980 | MR