Difference between revisions of "Independent functions, system of"
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| − | + | A sequence of measurable functions $ \{ f _ {i} \} $ | |
| + | such that | ||
| − | ( | + | $$ |
| + | \mu \{ {x } : {f _ {1} ( x) < | ||
| + | \alpha _ {1} \dots f _ {n} ( x) < \alpha _ {n} } \} | ||
| + | = \ | ||
| + | \prod _ {i=1} ^ { n } | ||
| + | \mu \{ {x } : {f _ {i} ( x) < \alpha _ {i} } \} | ||
| + | $$ | ||
| − | + | for any $ n $ | |
| + | and any $ \alpha _ {1} \dots \alpha _ {n} $. | ||
| + | The simplest example of a system of independent functions is the [[Rademacher system|Rademacher system]]. | ||
| − | + | (Kolmogorov's) criterion for the almost-everywhere convergence of a series of independent functions: For a series of independent functions $\sum_{i=1} ^ \infty f _ {i} $ | |
| + | to converge almost everywhere it is necessary and sufficient that for some $ C > 0 $ | ||
| + | the following three series converge: | ||
| + | |||
| + | $$ | ||
| + | \sum _ { i } \mu | ||
| + | \{ {x } : {f _ {i} ( x) > C } \} | ||
| + | ,\ \ | ||
| + | \sum _ { i } \int\limits | ||
| + | f _ {i} ^ { C } ( x) d x , | ||
| + | $$ | ||
| + | |||
| + | $$ | ||
| + | \sum _ { i } \int\limits ( f _ {i} ^ { C } ( x) ) ^ {2} \ | ||
| + | d x - \left ( \int\limits f _ {i} ^ { C } ( x) d x \right ) ^ {2} , | ||
| + | $$ | ||
where | where | ||
| − | + | $$ | |
| − | + | f _ {i} ^ { C } ( x) = \ | |
| + | \left \{ | ||
| + | \begin{array}{ll} | ||
| + | f _ {i} ( x) , & | f _ {i} ( x) | \leq C , \\ | ||
| + | 0 , & | f _ {i} ( x) | > C . \\ | ||
| + | \end{array} | ||
| + | \right .$$ | ||
====Comments==== | ====Comments==== | ||
| − | Of course, to be able to introduce the concept of a system of independent functions one needs to have a [[ | + | Of course, to be able to introduce the concept of a system of independent functions one needs to have a [[measure space]] $ ( X , \mu ) $ |
| + | on which the functions are defined and measurable (with respect to $ \mu $). | ||
| + | Moreover, $ \mu $ | ||
| + | must be positive and finite, so $ \mu $ | ||
| + | can be taken a [[Probability measure|probability measure]] (then $ ( X , \mu ) $ | ||
| + | is a [[Probability space|probability space]]). An example is $ ( X , \mu ) = ( [ 0 , 1 ], \textrm{ Lebesgue measure } ) $. | ||
In this abstract setting, instead of functions one takes random variables, thus obtaining a system of independent random variables. | In this abstract setting, instead of functions one takes random variables, thus obtaining a system of independent random variables. | ||
| − | The notion of a system of independent functions (random variables) should not be mixed up with that of an independent set of elements of a vector space | + | The notion of a system of independent functions (random variables) should not be mixed up with that of an independent set of elements of a vector space $ V $ |
| + | over a field $ K $: | ||
| + | A set of elements $ \{ x _ {1} \dots x _ {n} \} $ | ||
| + | in $ V $ | ||
| + | such that for $ c _ {i} \in K $, | ||
| + | $ c _ {1} x _ {1} + \dots + c _ {n} x _ {n} = 0 $ | ||
| + | implies $ c _ {1} = \dots = c _ {n} = 0 $, | ||
| + | see also [[Vector space|Vector space]]. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.-P. Kahane, "Some random series of functions" , Cambridge Univ. Press (1985)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.-P. Kahane, "Some random series of functions" , Cambridge Univ. Press (1985)</TD></TR></table> | ||
Latest revision as of 09:08, 6 January 2024
A sequence of measurable functions $ \{ f _ {i} \} $
such that
$$ \mu \{ {x } : {f _ {1} ( x) < \alpha _ {1} \dots f _ {n} ( x) < \alpha _ {n} } \} = \ \prod _ {i=1} ^ { n } \mu \{ {x } : {f _ {i} ( x) < \alpha _ {i} } \} $$
for any $ n $ and any $ \alpha _ {1} \dots \alpha _ {n} $. The simplest example of a system of independent functions is the Rademacher system.
(Kolmogorov's) criterion for the almost-everywhere convergence of a series of independent functions: For a series of independent functions $\sum_{i=1} ^ \infty f _ {i} $ to converge almost everywhere it is necessary and sufficient that for some $ C > 0 $ the following three series converge:
$$ \sum _ { i } \mu \{ {x } : {f _ {i} ( x) > C } \} ,\ \ \sum _ { i } \int\limits f _ {i} ^ { C } ( x) d x , $$
$$ \sum _ { i } \int\limits ( f _ {i} ^ { C } ( x) ) ^ {2} \ d x - \left ( \int\limits f _ {i} ^ { C } ( x) d x \right ) ^ {2} , $$
where
$$ f _ {i} ^ { C } ( x) = \ \left \{ \begin{array}{ll} f _ {i} ( x) , & | f _ {i} ( x) | \leq C , \\ 0 , & | f _ {i} ( x) | > C . \\ \end{array} \right .$$
Comments
Of course, to be able to introduce the concept of a system of independent functions one needs to have a measure space $ ( X , \mu ) $ on which the functions are defined and measurable (with respect to $ \mu $). Moreover, $ \mu $ must be positive and finite, so $ \mu $ can be taken a probability measure (then $ ( X , \mu ) $ is a probability space). An example is $ ( X , \mu ) = ( [ 0 , 1 ], \textrm{ Lebesgue measure } ) $.
In this abstract setting, instead of functions one takes random variables, thus obtaining a system of independent random variables.
The notion of a system of independent functions (random variables) should not be mixed up with that of an independent set of elements of a vector space $ V $ over a field $ K $: A set of elements $ \{ x _ {1} \dots x _ {n} \} $ in $ V $ such that for $ c _ {i} \in K $, $ c _ {1} x _ {1} + \dots + c _ {n} x _ {n} = 0 $ implies $ c _ {1} = \dots = c _ {n} = 0 $, see also Vector space.
References
| [a1] | J.-P. Kahane, "Some random series of functions" , Cambridge Univ. Press (1985) |
Independent functions, system of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Independent_functions,_system_of&oldid=13159