Lindeberg Condition
المؤلف:
Feller, W.
المصدر:
"Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung." Math. Zeit. 40
الجزء والصفحة:
...
24-4-2021
4077
Lindeberg Condition
A sufficient condition on the Lindeberg-Feller central limit theorem. Given random variates 
, 
, ..., let 
, the variance 
of 
be finite, and variance of the distribution consisting of a sum of 
s
 |
(1)
|
be
 |
(2)
|
In the terminology of Zabell (1995), let
 |
(3)
|
where 
denotes the expectation value of 
restricted to outcomes 
, then the Lindeberg condition is
 |
(4)
|
for all 
(Zabell 1995).
In the terminology of Feller (1971), the Lindeberg condition assumed that for each 
,
![1/(s_n^2)sum_(k=1)^nint_(|y|>=ts_n)y^2F_k<span style=]() {dy}->0, " src="https://mathworld.wolfram.com/images/equations/LindebergCondition/NumberedEquation5.gif" style="height:45px; width:170px" /> |
(5)
|
or equivalently
![1/(s_n^2)sum_(k=1)^nint_(|y|<ts_n)y^2F_k<span style=]() {dy}->1. " src="https://mathworld.wolfram.com/images/equations/LindebergCondition/NumberedEquation6.gif" style="height:45px; width:170px" /> |
(6)
|
Then the distribution
 |
(7)
|
tends to the normal distribution with zero expectation and unit variance (Feller 1971, p. 256). The Lindeberg condition (5) guarantees that the individual variances 
are small compared to their sum 
in the sense that for given 
for all sufficiently large 
, 
for 
, ..., 
(Feller 1971, p. 256).
REFERENCES:
Feller, W. "Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung." Math. Zeit. 40, 521-559, 1935.
Feller, W. "Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung, II." Math. Zeit. 42, 301-312, 1935.
Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, pp. 257-258, 1971.
Lindeberg, J. W. "Eine neue Herleitung des Exponential-gesetzes in der Wahrscheinlichkeitsrechnung." Math. Zeit. 15, 211-235, 1922.
Trotter, H. F. "An Elementary Proof of the Central Limit Theorem." Arch. Math. 10, 226-234, 1959.
Wallace, D. L. "Asymptotic Approximations to Distributions." Ann. Math. Stat. 29, 635-654, 1958.
Zabell, S. L. "Alan Turing and the Central Limit Theorem." Amer. Math. Monthly 102, 483-494, 1995.
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