40A05-KroneckersLemma.tex

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\pmtitle{Kronecker's lemma}
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\begin{document}
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.

\begin{lemma*}[Kronecker]
Let $x_1,x_2,\ldots$ and $0<b_1<b_2<\cdots$ 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, $b_n^{-1}\sum_{k=1}^nx_k\rightarrow 0$ as $n\rightarrow\infty$.  
\end{lemma*}
\begin{proof}
Set $u_n=\sum_{k=1}^nx_k/b_k$, 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
\begin{equation*}
\frac{a_{n+1}-a_n}{b_{n+1}-b_n}=u_n\rightarrow u_\infty
\end{equation*}
as $n\rightarrow\infty$. Then, the Stolz-Cesaro theorem says that $a_n/b_n$ also converges to $u_\infty$, so
\begin{equation*}
b_n^{-1}\sum_{k=1}^nx_k=b_n^{-1}\sum_{k=1}^nb_k(u_k-u_{k-1})=u_n-b_n^{-1}a_n\rightarrow 0.
\end{equation*}
\end{proof}

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