|
|
| (One intermediate revision by the same user not shown) |
| Line 1: |
Line 1: |
| 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]].
| | Hi there. Allow me begin by introducing the author, her name is Sophia Boon but she by no means really favored that name. Credit authorising is exactly where my main online psychic reading ([http://bigpolis.com/blogs/post/6503 bigpolis.com]) income comes from. Alaska is where I've usually been living. One of the issues she loves most [http://www.herandkingscounty.com/content/information-and-facts-you-must-know-about-hobbies clairvoyants] is canoeing and she's been performing it for fairly a whilst.<br><br>my web-site; real psychics, [http://www.aseandate.com/index.php?m=member_profile&p=profile&id=13352970 aseandate.com], |
| | |
| == 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 <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]]
| |
| | |
| {{mathanalysis-stub}}
| |
Hi there. Allow me begin by introducing the author, her name is Sophia Boon but she by no means really favored that name. Credit authorising is exactly where my main online psychic reading (bigpolis.com) income comes from. Alaska is where I've usually been living. One of the issues she loves most clairvoyants is canoeing and she's been performing it for fairly a whilst.
my web-site; real psychics, aseandate.com,