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In [[probability theory]], '''Kolmogorov's Three-Series Theorem''', named after [[Andrey Kolmogorov]], gives a criterion for the [[almost sure]] [[convergence of random variables|convergence]] of an [[infinite series]] of [[random variable]]s in terms of the convergence of three different series involving properties of their [[probability distribution]]s. Kolmogorov's three-series theorem, combined with [[Kronecker's lemma]], can be used to give a relatively easy proof of the [[Strong_Law_of_Large_Numbers#Strong_law|Strong Law of Large Numbers]].<ref>Durrett, Rick. "Probability: Theory and Examples." Duxbury advanced series, Third Edition, Thomson Brooks/Cole, 2005, Section 1.8, pp. 60-69.</ref>
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== Statement of the Theorem ==
 
 
Let (X<sub>n</sub>)<sub>n∈N</sub> be [[independent random variables]]. The random series ∑<sup>∞</sup><sub>n=1</sub>X<sub>n</sub> [[convergence of random variables|converges]] [[almost surely]] in ℝ if and only if the following conditions hold for some A > 0:
 
'''i.'''  ∑<sup>∞</sup><sub>n=1</sub>[[Image:U+2119.svg|9px]](|X|<sub>n</sub> &ge; A) converges
 
'''ii.'''  Let Y<sub>n</sub>:= X<sub>n</sub>1<sub>{|X|<sub>n</sub> &le; A}</sub>, then ∑<sup>∞</sup><sub>n=1</sub>&#120124;(Y<sub>n</sub>),  the series of [[expected value]]s of Y<sub>n</sub> , converges
 
'''iii.'''  ∑<sup>∞</sup><sub>n=1</sub>&#120141;ar(Y<sub>n</sub>) converges
 
== Proof ==
 
=== Sufficiency of Conditions ("if") ===
 
Condition (i) and [[Borel-Cantelli]] give that [[almost surely]] X<sub>n</sub> = Y<sub>n</sub> for n large, hence ∑<sup>∞</sup><sub>n=1</sub>X<sub>n</sub> converges if and only if ∑<sup>∞</sup><sub>n=1</sub>Y<sub>n</sub> converges. Conditions (ii)-(iii) and [[Kolmogorov's Two-Series Theorem]] given the almost sure convergence of ∑<sup>∞</sup><sub>n=1</sub>&#120124;(Y<sub>n</sub>).  
 
=== Necessity of Conditions ("only if") ===
Suppose that ∑<sup>∞</sup><sub>n=1</sub>X<sub>n</sub> converges almost surely.  
 
Without condition (i), by Borel-Cantelli there would exist some A > 0 such that almost surely {|X<sub>n</sub>| &le; A} for infinitely many values n, but then the series would diverge. Therefore we must have condition (i).  
 
We see that condition (iii) implies condition (ii): [[Kolmogorov's Two-Series Theorem]] along with condition (i) applied to the case A =1 gives the convergence of ∑<sub>n</sub>(Y<sub>n</sub> - &#120124;(Y<sub>n</sub>)). So given the convergence of ∑<sup>∞</sup><sub>n=1</sub>Y<sub>n</sub>, we have ∑<sup>∞</sup><sub>n=1</sub>&#120124;(Y<sub>n</sub>)  converges, so condition (ii) is implied.
 
Thus, it only remains to demonstrate the necessity of condition (iii), and we will have obtained the full result. It is equivalent to check condition (iii) for the series ∑<sup>∞</sup><sub>n=1</sub>Z<sub>n</sub> = ∑<sup>∞</sup><sub>n=1</sub>(Y<sub>n</sub> - Y'<sub>n</sub>) where for each n, Y<sub>n</sub> and Y'<sub>n</sub> are [[IID]]-- that is, to employ the assumption that  &#120124;(Y<sub>n</sub>) = 0, since Z<sub>n</sub> is a sequence of random variables bounded by 2, converging almost surely, and with &#120141;ar(Z<sub>n</sub>) = 2&#120141;ar(Y<sub>n</sub>). So we wish to check that if ∑<sup>∞</sup><sub>n=1</sub>Z<sub>n</sub> converges, ∑<sup>∞</sup><sub>n=1</sub>&#120141;ar(Z<sub>n</sub>) converges as well. This is a special case of a more general result from [[martingale theory]] with summands equal to the increments of a [[martingale]] sequence and the same conditions (&#120124;(Z<sub>n</sub>) = 0, series of the [[variance]]s converging, summands [[bounded]]).<ref>Sun, Rongfeng. Lecture notes. http://www.math.nus.edu.sg/~matsr/ProbI/Lecture4.pdf</ref><ref>M. Loève, "Probability theory", Princeton Univ. Press (1963) pp. Sect. 16.3</ref><ref>W. Feller, "An introduction to probability theory and its applications", 2, Wiley (1971) pp. Sect. IX.9</ref>
 
== Example ==
As an illustration of the theorem, consider the example of the ''[[Harmonic_series_(mathematics)#Random_harmonic_series|harmonic series with random signs]]'':
: <math> \sum_{n=1}^\infty \pm \frac{1}{n}.</math>
Here, "<math>\pm</math>" means that each term <math>1/n</math> is taken with a random sign that is either <math>1</math> or <math>-1</math> with respective probabilities <math>1/2,\ 1/2</math>, and all random signs are chosen independently. Letting <math>X_n</math> in the theorem denote a random variable that takes the values <math>1/n</math> and <math>-1/n</math> with equal probabilities, one can check easily that the conditions of the theorem are satisfied, so it follows that the harmonic series with random signs converges almost surely. On the other hand, the analogous series of (for example) square root reciprocals with random signs, namely
: <math> \sum_{n=1}^\infty \pm \frac{1}{\sqrt{n}},</math>
 
''diverges'' almost surely, since condition (3) in the theorem is not satisfied. Note that this is different from the behavior of the analogous series with ''alternating'' signs, <math>\sum_{n=1}^\infty (-1)^n/\sqrt{n}</math>, which does converge. In fact one can simply check that
: <math> \sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n}}=(\sqrt{2}-1)\zeta(\frac{1}{2})\approx -0.604899.</math>
 
==Notes==
<references/>
 
== External links ==
*{{cite web|title=Romik, Dan. Probability Theory Lecture Notes, Fall 2009, UC Davis.|url=http://www.math.ucdavis.edu/~romik/teaching/lectures2.pdf}}
 
[[Category:Mathematical series]]
[[Category:Probability theorems]]

Latest revision as of 15:27, 23 February 2014

The issue with the increasing perception that sees “knives” as weapons and seeks to make use of them in defence or aggression, is that this notion is self-fulfilling. Similarly, the desire to regulate knives, and even weapons, starts from the premise that their primary, and finally, solely use, is violence directed at different persons. MILWAUKEE, WI- Milwaukee Tool Company Automatic Pocket knife continues to develop their Hand Device providing with three new Fastback Knives, launching in November 2013. Increasing the utility knife line to incorporate pocket knives as well, the new additions to the Fastback household will include a Fastback Hawk Invoice Knife, Fastback Easy Pocket Knife and Fastback Spring Assisted Knife.

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This tiny pocket knife opens to reveal an impressive blade because of its good design. The cover folds all the best way around to kind a handle for the tucked away blade. It features a 440C stainless steel blade and a pointy zebra wooden and metal deal with. We like to think of the Douk-Douk Knife as the working man’s blade. It’s easy and hard. The flat design makes it the perfect travel companion when you need to toss a knife in your pocket, and the gunsmith-style ferro-blackened folded metal deal with provides a heaping spoonful of machismo to a easy and reasonably priced knife.

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These ultralight pocket knives are among the lightest on the market. The incredible ultralight pocket knives weigh in at only 34 grams. The all black modern knife has a 420B stainless-steel blade with a particular black titanium coating that has been microdented, which helps reduce via the toughest materials. Just because their name changed doesn't mean Schrade knives have turn out cheap automatic knives to be any less certified to be known as top-of-the-line knives in the marketplace. They have translated their unyielding quality to trendy manufacturing techniques, resulting in more accessible, superior knives that any of our nice grandparents would be proud to personal.

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Native legislation varies much more widely than state regulation relating to pocket knives. In Arizona, as an illustration, the state legislation allows the carry of pocket knives with blades of 4 inches and less. In Flagstaff and Florence, Arizona, nonetheless, chances are you'll solely carry knives with blades of three inches or much spring assisted knife less. Amongst native knife law it's common for bigger metropolis areas to decrease limits for legal blade length. In 1969, Queen Metropolis Cutlery was purchased by Servotronics Company who remains the guardian company of Queen Cutlery and its sister firm, the Ontario knife firm. To this present day, Queen knives are produced using old Schaat

Youngsters are typically fairly upfront and honest. If they need one thing they'll ask you for it. In case your son or daughter asks you for a pocket knife, then they are most definitely prepared. From there you'll, after all, want to assess the state of affairs. Do you are feeling that they're prepared? I do not have youngsters but I do understand how parents fear. Are you mature sufficient to be giving your child a pocket knife? A good query to ask yourself. You are welcome Anon. Many people grew up with pocket/pen knives as commonplace difficulty, this could help some of these folks to proceed with that custom.

At 3.75 ounces, the Spyderco Para-Army 2 makes a terrific every single day carry knife as well as a general use knife, and it does this because of the nice stability between weight, robustness and performance. The tip on the blade is a little bit thin for me to recommend it as a survival knife, but it will simply face up to any camping journey or daily carry operate you possibly can think of. Rep. Bennie Thompson of Mississipi authorized the decision to return small knives again into the prohibited gadgets record. He insinuates that “common sense” was sure to kick in after implementing such a reckless policy.

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