Kronecker’s lemma

Kronecker’s lemma gives a condition for convergence of partial sums of real numbers, and for example can be used in the proof of Kolmogorov’s strong law of large numbersMathworldPlanetmath.

Lemma (Kronecker).

Let x1,x2, and 0<b1<b2< be sequences of real numbers such that bn increases to infinityMathworldPlanetmath as n. Suppose that the sum n=1xn/bn convergesPlanetmathPlanetmath to a finite limit. Then, bn-1k=1nxk0 as n.


Set un=k=1nxk/bk, so that the limit u=limnun exists. Also set an=k=1n-1(bk+1-bk)uk so that


as n. Then, the Stolz-Cesaro theorem says that an/bn also converges to u, so


Title Kronecker’s lemma
Canonical name KroneckersLemma
Date of creation 2013-03-22 18:33:54
Last modified on 2013-03-22 18:33:54
Owner gel (22282)
Last modified by gel (22282)
Numerical id 6
Author gel (22282)
Entry type Theorem
Classification msc 40A05
Classification msc 40-00
Related topic StolzCesaroTheorem