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. ${\displaystyle (a_{k})_{k\in \mathbb {N} }}$

Let be a monotonic, unbounded sequence of positive real numbers. ${\displaystyle (b_{k})_{k\in \mathbb {N} }}$

If converges, it follows . ${\displaystyle \sum _{k=1}^{n}{\frac {a_{k}}{b_{k}}}}$ ${\displaystyle {\frac {1}{b_{n}}}\sum _{k=1}^{n}a_{k}\to 0}$

Inference

The above lemma simplifies to the following statement when setting for all : ${\displaystyle b_{k}=k}$ ${\displaystyle k\in \mathbb {N} }$

Be a sequence of real numbers. ${\displaystyle (a_{k})_{k\in \mathbb {N} }}$

If converges, it follows . ${\displaystyle \sum _{k=1}^{n}{\frac {a_{k}}{k}}}$ ${\displaystyle {\frac {1}{n}}\sum _{k=1}^{n}a_{k}\to 0}$

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