Scheme of random variables

In probability theory, a scheme of random variables , also known as a triangle scheme , denotes a generalization of a sequence of random variables in which the random variables are combined into smaller groups using a second index. This has the advantage that certain properties (normalization, centeredness, independence ) only have to be required for these subgroups and not for the entire sequence and yet certain statements can still be made. Schemes of random variables play a role in Lindeberg-Feller's central limit theorem , a generalization of the central limit theorem for partial sums of schemes of random variables.

definition

For each one is given and random variables . Then is called ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle k_{n}\in \mathbb {N} }$ ${\displaystyle X_{n,1},\dots ,X_{n,k_{n}}}$

${\displaystyle (X_{n,l})=(X_{n,l},\,l=1,\dots ,k_{n}{\text{ für }}n\in \mathbb {N} )}$

a scheme of random variables . The random variables are therefore always grouped in size groups. ${\displaystyle k_{n}}$

example

A sequence of independently identically distributed random variables is given . We put for the sake of simplicity for everyone . The groups therefore all consist of 4 random variables. The schema is now defined as ${\displaystyle (Y_{n})_{n\in \mathbb {N} }}$ ${\displaystyle k_{n}=4}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle X_{n,l}=\sum _{i=n(l-1)+1}^{nl}Y_{i}}$.

Partial sums of the sequences are therefore always formed which have the length and do not mutually overlap. Written out the scheme would look like this: ${\displaystyle n}$

${\displaystyle {\begin{matrix}&l=1&l=2&l=3&l=4\\n=1&Y_{1}&Y_{2}&Y_{3}&Y_{4}\\n=2&Y_{1}+Y_{2}&Y_{3}+Y_{4}&Y_{5}+Y_{6}&Y_{7}+Y_{8}\\n=3&Y_{1}+Y_{2}+Y_{3}&Y_{4}+Y_{5}+Y_{6}&Y_{7}+Y_{8}+Y_{9}&Y_{10}+Y_{11}+Y_{12}\end{matrix}}}$

Clarifications

As with sequences of random variables, some more precise definitions of the term can be given for schemes of random variables.

Independent scheme

A scheme of random variables is called an independent scheme if for all the random variables are stochastically independent . ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle (X_{n,l})_{l=1,\dots ,k_{n}}}$

Centered scheme

A scheme of random variables is called a centered scheme , if is for all . ${\displaystyle \operatorname {E} (X_{n,l})=0}$ ${\displaystyle n,l}$

Standardized scheme

A scheme of random variables is called a normalized scheme if that applies to all of them ${\displaystyle n\in \mathbb {N} }$

${\displaystyle \sum _{l=1}^{k_{n}}\operatorname {Var} (X_{n,l})=1}$

is.

Asymptotically negligible scheme

A centered scheme of random variables is called an asymptotically negligible scheme if

${\displaystyle \lim _{n\to \infty }\max _{1\leq l\leq k_{n}}P(|X_{n,l}|>\varepsilon )=0}$

is for everyone . ${\displaystyle \epsilon >0}$

Examples

• The scheme considered above is an independent scheme, because partial sums of sequences of independent random variables that have no summands in common are again independent. If the expectation value is 0, then all the partial sums also have the expectation value 0 and this is also a centered schema. Without further information about the random variables, nothing can be said about normalization or asymptotic negligibility. ${\displaystyle Y_{1}}$
• If a sequence of independently identically distributed random variables with a common expectation and variance is through ${\displaystyle (Y_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \operatorname {E} (Y_{n})=\mu }$ ${\displaystyle \operatorname {Var} (Y_{n})=\sigma ^{2}}$

${\displaystyle X_{n,l}={\frac {Y_{l}-\mu }{\sigma {\sqrt {n}}}}}$

with , an independent, centered, standardized and asymptotically negligible scheme is given. ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle l=1,\dotsc ,n}$

Web links

• AV Prokhorov: Asymptotic negligibility . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

literature

• 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 .
• 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 .