Local limit sets

As a local limit theorems to certain designated mathematical theorems that the limit theorems stochastics are counted. Like all of these limit sets, the local limit sets examine sequences and sums of random variables . In contrast to these, however, they do not use the classic convergence terms of stochastics such as convergence in distribution , convergence in probability or almost certain convergence , but investigate the convergence of probability functions and probability density functions .

Task

A sequence of random variables with distributions and probability functions or probability density functions is given . We are looking for a probability (density) function and conditions under which ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle P_{n}}$ ${\displaystyle f_{n}}$ ${\displaystyle f}$

${\displaystyle f_{n}}$against converges. ${\displaystyle f}$

Possible problems are

• Generally, the limit function must even with convergence in distribution of do not have density function. ${\displaystyle X_{n}}$
• Even if the random variables in distribution converge to the standard normal distribution, the densities generally do not have to converge.

Example: De Moivre-Laplace local limit theorem

A classic example is de Moivre-Laplace's local limit theorem . Is , let be the probability density function of the standard normal distribution and ${\displaystyle p\in (0,1)}$ ${\displaystyle \varphi }$

${\displaystyle x_{n}(k)={\frac {k-np}{\sqrt {np(1-p)}}}}$.

Then is for any ${\displaystyle c>0}$

${\displaystyle \lim _{n\to \infty }\max _{\forall k \atop |x_{n}(k)|\leq c}\left|{\frac {{\sqrt {np(1-p)}}\operatorname {Bin} _{n,p}(\{k\})}{\varphi (x_{n}(k))}}-1\right|=0}$.

Web links

• VV Petrov: Local limit theorems . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• AV Prokhorov: Laplace theorem . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

literature

• Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , doi : 10.1515 / 9783110215274 .