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| In [[mathematics]], '''Kronecker's lemma''' (see, e.g., {{harvtxt|Shiryaev|1996|loc=Lemma IV.3.2}}) is a result about the relationship between convergence of [[infinite sum]]s and convergence of sequences. The lemma is often used in the proofs of theorems concerning sums of independent random variables such as the strong [[Law of large numbers]]. The lemma is named after the [[Germany|German]] [[mathematician]] [[Leopold Kronecker]].
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| == The lemma ==
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| If <math>(x_n)_{n=1}^\infty</math> is an infinite sequence of real numbers such that
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| :<math>\sum_{m=1}^\infty x_m = s</math>
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| exists and is finite, then we have for <math>0<b_1 \leq b_2 \leq b_3 \leq \ldots</math> and <math>b_n \to \infty</math> that
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| :<math>\lim_{n \to \infty}\frac1{b_n}\sum_{k=1}^n b_kx_k = 0.</math>
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| ===Proof===
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| Let <math>S_k</math> denote the partial sums of the ''x'''s. Using [[summation by parts]],
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| : <math>\frac1{b_n}\sum_{k=1}^n b_k x_k = S_n - \frac1{b_n}\sum_{k=1}^{n-1}(b_{k+1} - b_k)S_k</math> | |
| Pick any ''ε'' > 0. Now choose ''N'' so that <math>S_k</math> is ''ε''-close to ''s'' for ''k'' > ''N''. This can be done as the sequence <math>S_k</math> converges to ''s''. Then the right hand side is:
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| : <math>S_n - \frac1{b_n}\sum_{k=1}^{N-1}(b_{k+1} - b_k)S_k - \frac1{b_n}\sum_{k=N}^{n-1}(b_{k+1} - b_k)S_k</math>
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| : <math>= S_n - \frac1{b_n}\sum_{k=1}^{N-1}(b_{k+1} - b_k)S_k - \frac1{b_n}\sum_{k=N}^{n-1}(b_{k+1} - b_k)s - \frac1{b_n}\sum_{k=N}^{n-1}(b_{k+1} - b_k)(S_k - s)</math>
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| : <math>= S_n - \frac1{b_n}\sum_{k=1}^{N-1}(b_{k+1} - b_k)S_k - \frac{b_n-b_N}{b_n}s - \frac1{b_n}\sum_{k=N}^{n-1}(b_{k+1} - b_k)(S_k - s).</math>
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| Now, let ''n'' go to infinity. The first term goes to ''s'', which cancels with the third term. The second term goes to zero (as the sum is a fixed value). Since the ''b'' sequence is increasing, the last term is bounded by <math>\epsilon (b_n - b_N)/b_n \leq \epsilon</math>.
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| ==References==
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| {{refbegin}}
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| * {{cite book
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| | last = Shiryaev | first = Albert N.
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| | title = Probability
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| | year = 1996
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| | edition = 2nd
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| | publisher = Springer
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| | isbn = 0-387-94549-0
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| | ref = CITEREFShiryaev1996
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| }}
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| {{refend}}
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| [[Category:Mathematical series]]
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| [[Category:Lemmas]]
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| {{mathanalysis-stub}}
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I am Oscar and I totally dig that name. To gather coins is what his family members and him enjoy. Hiring has been my profession for some time but I've currently applied for an additional one. North Dakota is her beginning location but she will have to transfer 1 day or an additional.
My weblog :: at home std test