View source for Lindeberg-Feller theorem

A theorem that establishes necessary and sufficient conditions for the asymptotic normality of the distribution function of sums of independent random variables that have finite variances. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058940/l0589401.png" /> be a sequence of independent random variables with means <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058940/l0589402.png" /> and finite variances <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058940/l0589403.png" /> not all of which are zero. Let

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In order that

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and

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for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058940/l0589407.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058940/l0589408.png" />, it is necessary and sufficient that the following condition (the Lindeberg condition) is satisfied:

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as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058940/l05894010.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058940/l05894011.png" />. Sufficiency was proved by J.W. Lindeberg [[#References|[1]]] and necessity by W. Feller [[#References|[2]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.W. Lindeberg,   "Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung"  ''Math. Z.'' , '''15'''  (1922)  pp. 211–225</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Feller,   "Ueber den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung"  ''Math. Z.'' , '''40'''  (1935)  pp. 521–559</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Loève,   "Probability theory" , Springer  (1977)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.V. Petrov,   "Sums of independent random variables" , Springer  (1975)  (Translated from Russian)</TD></TR></table>