Lyapunov condition

In stochastics, the Lyapunov condition is a criterion for a sequence of random variables . In addition to the more general Lindeberg condition, it is one of the two classic sufficient prerequisites for the convergence in the distribution of the sequence against the standard normal distribution and thus belongs to the topic of the central limit theorems . It can also be formulated for schemes of random variables and goes back to the Russian mathematician Alexander Mikhailovich Lyapunov .

Formulation for sequences of random variables

Let be a sequence of stochastically independent random variables with ${\displaystyle (X_{i})_{i\in \mathbb {N} }}$

${\displaystyle \;a_{i}:=\operatorname {E} [X_{i}]}$and for everyone . ${\displaystyle 0<\operatorname {Var} (X_{i})<\infty }$ ${\displaystyle i\in \mathbb {N} }$

The random variables can also have different distributions. Also denote

${\displaystyle S_{n}=\sum _{i=1}^{n}\operatorname {Var} (X_{i})}$

the sum of the variances of the . ${\displaystyle (X_{i})_{i}}$

The sequence of random variables now satisfies Lyapunov's condition if and only if there exists such that ${\displaystyle \delta >0}$

${\displaystyle \lim _{n\rightarrow \infty }{\frac {1}{S_{n}^{1+\delta /2}}}\sum _{i=1}^{n}E\left[|X_{i}-a_{i}|^{2+\delta }\right]=0}$

applies.

Formulation for schemes of random variables

Let an independent centered scheme of random variables be given , in which every random variable is and are square-integrable ${\displaystyle (X_{n,l})}$ ${\displaystyle X_{n,l}}$

${\displaystyle S_{n}:=\sum _{l=1}^{k_{n}}X_{n,l}}$

the sums over the second indices. The schema now satisfies the Lyapunov condition if one exists such that ${\displaystyle \delta >0}$

${\displaystyle \lim _{n\to \infty }{\frac {1}{\operatorname {Var} (S_{n})^{1+\delta /2}}}\sum _{l=1}^{k_{n}}\operatorname {E} (|X_{n,l}|^{2+\delta })=0}$

is.

Relation to the Lindeberg condition

The Lyapunov condition always implies the Lindeberg condition , but the reverse is generally not true. Therefore it is dealt with more often in the literature.

Lyapunov's theorem

The statement that the Lyapunov condition is sufficient for the convergence in distribution to the standard normal distribution is referred to in the literature as Lyapunov's theorem or Lyapunov's central limit theorem . In full it reads:

If a sequence of stochastically independent real-valued random variables with finite second moments suffices for Lyapunov's condition, then the rescaled sequence of the centered random variables in distribution converges to the standard normal distribution: ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$

${\displaystyle \lim _{n\rightarrow \infty }{\frac {\sum _{i=1}^{n}(X_{i}-\operatorname {E} (X_{i}))}{\sqrt {\sum _{i=1}^{n}\operatorname {Var} (X_{i})}}}\;{\overset {d}{\longrightarrow }}\;{\mathcal {N}}(0,1)}$

It was shown by Alexander Michailowitsch Lyapunow in 1901 and generalized to Jarl Waldemar Lindeberg in 1922 by the Lindeberg theorem , which is based on the Lindeberg condition.

literature

• Heinz Bauer : Probability Theory. De Gruyter, Berlin 2002, ISBN 3-11-017236-4 .
• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
• Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , doi : 10.1007 / 978-3-642-45387-8 .
• David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , doi : 10.1007 / b137972 .

Individual evidence

1. ↑ Meintrup, Schäffler: Stochastics. 2005, p. 204.
2. ↑ Klenke: Probability Theory. 2013, p. 327.
3. ↑ Eric W. Weisstein : Lyapunov Condition . In: MathWorld (English).
4. ^ AV Prokhorov: Lyapunov theorem . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
5. ↑ Kusolitsch: Measure and probability theory. 2014, p. 307.