Geodesic
On the Necessity of the Lindeberg Condition for the Normal Convergence of Martingales
Matematičeskie zametki, Tome 70 (2001) no. 1, pp. 51-58.

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We give conditions on square-integrable martingales whose consequence is the “conditional” Lindeberg condition. 
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V. M. Kruglov. On the Necessity of the Lindeberg Condition for the Normal Convergence of Martingales. Matematičeskie zametki, Tome 70 (2001) no. 1, pp. 51-58. http://geodesic.mathdoc.fr/item/MZM_2001_70_1_a6/

[1] Shiryaev A. N., Veroyatnost, 2-e izd., Nauka, M., 1989

[2] Sholomitskii A. G., “O neobkhodimykh usloviyakh normalnoi skhodimosti dlya martingalov”, Teor. veroyatn. i ee prim., 43:3 (1998), 490–508 | MR

[3] Loev M., Teoriya veroyatnostei, IL, M., 1962