Lindeberg's condition

Let $(\backslash Omega,\; \backslash mathcal\{F\},\; \backslash mathbb\{P\})$ be a probability space, and $X\_k\; :\; \backslash Omega\; \backslash to\; \backslash mathbb\{R\},\backslash ,\backslash ,\; k\; \backslash in\; \backslash mathbb\{N\}$, be independent random variables defined on that space. Assume the expected values $\backslash mathbb\{E\}\backslash ,[X\_k]\; =\; \backslash mu\_k$ and variances $\backslash mathrm\{Var\}\backslash ,[X\_k]\; =\; \backslash sigma\_k^2$ exist and are finite. Also let $s\_n^2\; :=\; \backslash sum\_\{k=1\}^n\; \backslash sigma\_k^2\; .$

If this sequence of independent random variables $X\_k$ satisfies Lindeberg's condition:

:$\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash frac\{1\}\{s\_n^2\}\backslash sum\_\{k\; =\; 1\}^n\; \backslash mathbb\{E\}\; \backslash left[(X\_k\; -\; \backslash mu\_k)^2\; \backslash cdot\; \backslash mathbf\{1\}\_\{\backslash \{\; |\; X\_k\; -\; \backslash mu\_k\; |\; >\; \backslash varepsilon\; s\_n\; \backslash \}\}\; \backslash right]\; =\; 0$

for all $\backslash varepsilon\; >\; 0$, where 1_{…} is the indicator function, then the central limit theorem holds, i.e. the random variables

:$Z\_n\; :=\; \backslash frac\{\backslash sum\_\{k\; =\; 1\}^n\; (X\_k\; -\; \backslash mu\_k)\}\{s\_n\}$

converge in distribution to a standard normal random variable as $n\; \backslash to\; \backslash infty.$

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

:$\backslash max\_\{k=1,\backslash ldots,n\}\; \backslash frac\{\backslash sigma\_k^2\}\{s\_n^2\}\; \backslash to\; 0,\; \backslash quad\; \backslash text\{\; as\; \}\; n\; \backslash to\; \backslash infty,$

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

Feller's theorem can be used as an alternative method to prove that Lindeberg's condition holds.{{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 }} Letting $S\_n\; :=\; \backslash sum\_\{k=1\}^n\; X\_k$ and for simplicity $\backslash mathbb\{E\}\backslash ,[X\_k]\; =\; 0$, the theorem states

:if $\backslash forall\; \backslash varepsilon\; >\; 0$, $\backslash lim\_\{n\; \backslash rightarrow\; \backslash infty\}\; \backslash max\_\{1\; \backslash leq\; k\; \backslash leq\; n\}\; P(|X\_k|\; >\; \backslash varepsilon\; s\_n)\; =\; 0$ and $\backslash frac\{S\_n\}\{s\_n\}$ converges weakly to a standard normal distribution as $n\; \backslash rightarrow\; \backslash infty$ then $X\_k$ satisfies the Lindeberg's condition.

This theorem can be used to disprove the central limit theorem holds for $X\_k$ by using proof by contradiction. This procedure involves proving that Lindeberg's condition fails for $X\_k$.

Because the Lindeberg condition implies $\backslash max\_\{k=1,\backslash ldots,n\}\backslash frac\{\backslash sigma^2\_k\}\{s\_n^2\}\; \backslash to\; 0$ as $n\; \backslash to\; \backslash infty$, it guarantees that the contribution of any individual random variable $X\_k$ ($1\backslash leq\; k\backslash leq\; n$) to the variance $s\_n^2$ is arbitrarily small, for sufficiently large values of $n$.

Consider the following informative example which satisfies the Lindeberg condition. Let $\backslash xi\_i$ be a sequence of zero mean, variance 1 iid random variables and $a\_i$ a non-random sequence satisfying:

$$\backslash max\_i^n\; \backslash frac$$

Now, define the normalized elements of the linear combination:

$$X\_\{i\}\; =\; \backslash frac\{a\_i\; \backslash xi\_i\}\{\backslash |a\backslash |\_2\}$$

which satisfies the Lindeberg condition:

$$\backslash sum\_\{i=1\}^n\; \backslash mathbb\; E\; \backslash left\; [\; \backslash left\; |\; X\_i\backslash right\; |^2\; 1(|X\_i|\; >\; \backslash varepsilon)\backslash right\; ]\; \backslash leq\; \backslash sum\_\{i=1\}^n\; \backslash mathbb\; E\; \backslash left\; [\; \backslash left\; |\; X\_i\backslash right\; |^2\; 1\; \backslash left(|\backslash xi\_i|\; >\; \backslash varepsilon\; \backslash frac\{\backslash |a\backslash |\_2\}\{\backslash max\_i^n\; |a\_i|\}\; \backslash right)\backslash right\; ]\; =\; \backslash sum\_\{i=1\}^\{n\}$$

\mathbb E \left [ \left | \xi_i\right |^2 1 \left(|\xi_i| > \varepsilon \frac{\|a\|_2}{\max_i^n |a_i|} \right)\right ]

but $\backslash xi\_i^2$ is finite so by DCT and the condition on the $a\_i$ we have that this goes to 0 for every $\backslash varepsilon$.

• Lyapunov condition
• Central limit theorem

{{Reflist}}

Category:Theorems in statistics

Category:Central limit theorem