# 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 numbers.

###### Lemma (Kronecker).

Let $x_{1},x_{2},\ldots$ and $0 be sequences of real numbers such that $b_{n}$ increases to infinity as $n\rightarrow\infty$. Suppose that the sum $\sum_{n=1}^{\infty}x_{n}/b_{n}$ converges to a finite limit. Then, $b_{n}^{-1}\sum_{k=1}^{n}x_{k}\rightarrow 0$ as $n\rightarrow\infty$.

###### Proof.

Set $u_{n}=\sum_{k=1}^{n}x_{k}/b_{k}$, so that the limit $u_{\infty}=\lim_{n\rightarrow\infty}u_{n}$ exists. Also set $a_{n}=\sum_{k=1}^{n-1}(b_{k+1}-b_{k})u_{k}$ so that

 $\frac{a_{n+1}-a_{n}}{b_{n+1}-b_{n}}=u_{n}\rightarrow u_{\infty}$

as $n\rightarrow\infty$. Then, the Stolz-Cesaro theorem says that $a_{n}/b_{n}$ also converges to $u_{\infty}$, so

 $b_{n}^{-1}\sum_{k=1}^{n}x_{k}=b_{n}^{-1}\sum_{k=1}^{n}b_{k}(u_{k}-u_{k-1})=u_{% n}-b_{n}^{-1}a_{n}\rightarrow 0.$

Title Kronecker’s lemma KroneckersLemma 2013-03-22 18:33:54 2013-03-22 18:33:54 gel (22282) gel (22282) 6 gel (22282) Theorem msc 40A05 msc 40-00 StolzCesaroTheorem