Kronecker's lemma

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In mathematics, Kronecker's lemma is a result about the relationship between convergence of infinite sums and convergence of sequences. The lemma is often used as part of the proofs concerning theorems concerning sums of independent random variables such as the strong Law of large numbers. The lemma is named after the German mathematician Leopold Kronecker.

[edit] The lemma

If (x_n)_{n=1}^\infty is an infinite sequence of real numbers such that

\sum_{n=1}^\infty x_n = s

exists and is finite, then we have for 0<b_1 \leq b_2 \leq b_3 \leq \ldots and b_n \to \infty that

\lim_{n \to \infty}\frac1{b_n}\sum_{k=1}^n b_kx_k = 0

(Proof)


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