convergence in the mean



be the arithmetic meanMathworldPlanetmath of the numbers a1,a2,,an.   The sequenceMathworldPlanetmath

a1,a2,a3, (1)

is said to converge in the mean ( iff the sequence

b1,b2,b3, (2)

On has the

Theorem.  If the sequence (1) is convergentMathworldPlanetmathPlanetmath having the limit A, then also the sequence (2) converges to the limit A.  Thus, a convergent sequence is always convergent in the mean.

Proof.  Let ε be an arbitrary positive number.  We may write

|A-bn| =|A-1n(a1++ak)-1n(ak+1++an)|

The supposition implies that there is a positive integer k such that

|A-ai|<ε2 for all i>k.

Let’s fix the integer k.  Choose the number l so great that

|(A-a1)++(A-ak)|n<ε2 for n>l.

Let now  n>max{k,l}.  The three above inequalitiesMathworldPlanetmath yield


whence we have


Note.  The converse ( of the theorem is not true.  For example, if


i.e. if the sequence (1) has the form  0,1,0,1,0,1,,  then it is divergent but converges in the mean to the limit 12; the corresponding sequence (2) is 0,12,13,24,25,36,37,48,49,

Title convergence in the mean
Canonical name ConvergenceInTheMean
Date of creation 2015-04-08 7:29:35
Last modified on 2015-04-08 7:29:35
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Definition