Kronecker's lemma
The Kronecker lemma deals with limits in mathematics. It is named after the German mathematician Leopold Kronecker .
lemma
Be a sequence of real numbers.
Let be a monotonic, unbounded sequence of positive real numbers.
If converges, it follows .
Inference
The above lemma simplifies to the following statement when setting for all :
Be a sequence of real numbers.
If converges, it follows .
application
Kronecker's lemma can be used to prove the strong law of large numbers .
literature
- Albrecht Irle : Probability Theory and Statistics: Basics - Results - Applications . 2nd Edition. Vieweg + Teubner, 2005, ISBN 978-3-519-12395-8 . Pages 190 and 194
- Acta Mathematica Hungarica, Volume 44, Numbers 1-2, March 1984, pages 143 and 144