Independent functions, system of

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)