View source for Kronecker's lemma

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]].

== The lemma ==
If <math>(x_n)_{n=1}^\infty</math> is an infinite sequence of real numbers such that 
:<math>\sum_{m=1}^\infty x_m = s</math> 
exists and is finite, then we have for all <math>0<b_1 \leq b_2 \leq b_3 \leq \ldots</math> and <math>b_n \to \infty</math> that 
:<math>\lim_{n \to \infty}\frac1{b_n}\sum_{k=1}^n b_kx_k = 0.</math>

===Proof===
Let <math>S_k</math> denote the partial sums of the ''x'''s. Using [[summation by parts]],
: <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:
: <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>
: <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>
: <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>
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>.

==References==
{{refbegin}}
* {{cite book
  | last = Shiryaev | first = Albert N.
  | title = Probability
  | year = 1996
  | edition = 2nd
  | publisher = Springer
  | isbn = 0-387-94549-0
  | ref = CITEREFShiryaev1996
  }}
{{refend}}

[[Category:Mathematical series]]
[[Category:Lemmas]]

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