independent sigma algebras
An arbitrary set of sub--algebras is mutually independent if any finite subset of is independent.
Events (in ) are mutually independent if the sigma algebras are mutually independent.
Random variables defined on are mutually independent if the sigma algebras generated by (http://planetmath.org/MathcalFMeasurableFunction) the ’s are mutually independent.
In general, mutual independence among events , random variables , and sigma algebras means the mutual independence among , , and .
Remark. Even when random variables are defined on different probability spaces , we may form the product (http://planetmath.org/InfiniteProductMeasure) of these spaces so that (by abuse of notation) are now defined on and their independence can be discussed.
|Title||independent sigma algebras|
|Date of creation||2013-03-22 16:22:58|
|Last modified on||2013-03-22 16:22:58|
|Last modified by||CWoo (3771)|
|Synonym||mutually independent -algebras|
|Defines||mutually independent sigma algebras|