Lindeberg's condition

{{Short description|Theorem from probability theory}} In probability theory, '''Lindeberg's condition''' is a sufficient condition (and under certain conditions also a necessary condition) for the central limit theorem (CLT) to hold for a sequence of independent random variables.<ref>{{cite book |first=P. |last=Billingsley |title=Probability and Measure |publisher=Wiley |year=1986 |edition=2nd |page=369 |isbn=0-471-80478-9 |url=https://books.google.com/books?id=Q2IPAQAAMAAJ&pg=PA369 }}</ref><ref>{{cite book |first=R. B. |last=Ash |title=Probability and measure theory |url=https://archive.org/details/probabilitymeasu00ashr_983 |url-access=limited |year=2000 |edition=2nd |page=[https://archive.org/details/probabilitymeasu00ashr_983/page/n324 307] |isbn=0-12-065202-1 }}</ref><ref>{{cite book |first=S. I. |last=Resnick | title=A probability Path |url=https://archive.org/details/probabilitypath00resn_981 |url-access=limited |year=1999 |page=[https://archive.org/details/probabilitypath00resn_981/page/n322 314] }}</ref> Unlike the classical CLT, which requires that the random variables in question have finite variance and be both independent and identically distributed, Lindeberg's CLT only requires that they have finite variance, satisfy Lindeberg's condition, and be independent. It is named after the Finnish mathematician Jarl Waldemar Lindeberg.<ref>{{cite journal |first=J. W. |last=Lindeberg | title=Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung | year=1922 | pages=211–225 | journal = Mathematische Zeitschrift |volume = 15 | issue = 1 | doi = 10.1007/BF01494395 |s2cid=119730242 | url=http://gdz.sub.uni-goettingen.de/en/dms/load/img/?PPN=PPN266833020_0015&DMDID=dmdlog21}}</ref>

==Statement==

Let <math>(\Omega, \mathcal{F}, \mathbb{P})</math> be a probability space, and <math>X_k : \Omega \to \mathbb{R},\,\, k \in \mathbb{N}</math>, be ''independent'' random variables defined on that space. Assume the expected values <math>\mathbb{E}\,[X_k] = \mu_k</math> and variances <math>\mathrm{Var}\,[X_k] = \sigma_k^2</math> exist and are finite. Also let <math>s_n^2 := \sum_{k=1}^n \sigma_k^2 .</math>

If this sequence of independent random variables <math>X_k</math> satisfies '''Lindeberg's condition''':

:<math> \lim_{n \to \infty} \frac{1}{s_n^2}\sum_{k = 1}^n \mathbb{E} \left[(X_k - \mu_k)^2 \cdot \mathbf{1}_{\{ | X_k - \mu_k | > \varepsilon s_n \}} \right] = 0</math>

for all <math>\varepsilon > 0</math>, where '''1'''<sub>{…}</sub> is the indicator function, then the central limit theorem holds, i.e. the random variables

:<math>Z_n := \frac{\sum_{k = 1}^n (X_k - \mu_k)}{s_n}</math>

converge in distribution to a standard normal random variable as <math>n \to \infty.</math>

Lindeberg's condition is sufficient, but not in general necessary (i.e. the inverse implication does not hold in general). However, if the sequence of independent random variables in question satisfies

:<math>\max_{k=1,\ldots,n} \frac{\sigma_k^2}{s_n^2} \to 0, \quad \text{ as } n \to \infty,</math>

then Lindeberg's condition is both sufficient and necessary, i.e. it holds if and only if the result of central limit theorem holds.

==Remarks== ===Feller's theorem=== Feller's theorem can be used as an alternative method to prove that Lindeberg's condition holds.<ref>{{cite book |first=K. B. |last=Athreya |first2=S. N. |last2=Lahiri |year=2006 |title=Measure Theory and Probability Theory |location= |publisher=Springer |page=348 |isbn=0-387-32903-X }}</ref> Letting <math>S_n := \sum_{k=1}^n X_k</math> and for simplicity <math>\mathbb{E}\,[X_k] = 0</math>, the theorem states

:if <math>\forall \varepsilon > 0 </math>, <math>\lim_{n \rightarrow \infty} \max_{1 \leq k \leq n} P(|X_k| > \varepsilon s_n) = 0</math> and <math>\frac{S_n}{s_n}</math> converges weakly to a standard normal distribution as <math>n \rightarrow \infty</math> then <math>X_k</math> satisfies the Lindeberg's condition.

This theorem can be used to disprove the central limit theorem holds for <math>X_k</math> by using proof by contradiction. This procedure involves proving that Lindeberg's condition fails for <math>X_k</math>.

==Interpretation==

Because the Lindeberg condition implies <math>\max_{k=1,\ldots,n}\frac{\sigma^2_k}{s_n^2} \to 0</math> as <math>n \to \infty</math>, it guarantees that the contribution of any individual random variable <math>X_k</math> (<math>1\leq k\leq n</math>) to the variance <math>s_n^2</math> is arbitrarily small, for sufficiently large values of <math>n</math>.

=== Example === Consider the following informative example which satisfies the Lindeberg condition. Let <math>\xi_i</math> be a sequence of zero mean, variance 1 iid random variables and <math>a_i</math> a non-random sequence satisfying:

<math display=block> \max_i^n \frac{|a_i|}{\|a\|_2} \rightarrow 0</math>

Now, define the normalized elements of the linear combination:

which satisfies the Lindeberg condition:

<math display=block> \sum_{i=1}^n \mathbb E \left [ \left | X_i\right |^2 1(|X_i| > \varepsilon)\right ] \leq \sum_{i=1}^n \mathbb E \left [ \left | X_i\right |^2 1 \left(|\xi_i| > \varepsilon \frac{\|a\|_2}{\max_i^n |a_i|} \right)\right ] = \sum_{i=1}^{n} \mathbb E \left [ \left | \xi_i\right |^2 1 \left(|\xi_i| > \varepsilon \frac{\|a\|_2}{\max_i^n |a_i|} \right)\right ]</math>

but <math>\xi_i^2</math> is finite so by DCT and the condition on the <math>a_i</math> we have that this goes to 0 for every <math>\varepsilon</math>.

==References== {{Reflist}}

Category:Theorems in statistics Category:Central limit theorem