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	~~Hello buddy~~. ~~Allow me introduce myself. I am Ron but I don~~'~~t like when individuals use my full title~~. ~~The~~ [~~http://www~~.~~Autobytel.com~~/~~car~~-~~buying-tips~~/~~warranty~~-~~information/ favorite pastime] for my children~~ and me is ~~taking part in crochet and now~~ [~~https://Autoclubsouth~~.~~Aaa.com~~/~~Automotive~~/~~auto_warranty.aspx?nvbar=Automotive~~:~~ExtendedWarranty I'm attempting~~] [~~http:~~//~~www~~.~~chillister.com~~/~~blogs~~/~~post/27388 auto warranty]~~ to ~~make extended [http:~~//89.1.13.~~189~~/~~index.php?mod~~=~~users&action~~=~~view&id~~=~~63366 car warranty] cash with it~~. ~~Managing people~~ is ~~how I make cash and~~ it~~'s something I really appreciate. Her spouse and her extended car warranty selected~~ to ~~reside in Delaware~~ but ~~she requirements~~ to ~~move simply extended auto [http~~://~~Autos~~.~~Jdpower~~.~~com/content/buying-tip/xa345A7/pros-~~and~~-cons-~~of~~-buying-an-extended-warranty-for-~~a~~-car.htm warranty] because of her family members~~.<br><br>~~my web blog~~ ... ~~[http://xn--h1aecfj6f~~.xn--~~p1ai/content/auto~~-~~repair-tips-youll-wish-youd-read-sooner ильком.xn--p1ai~~]		The '''Hewitt–Savage zero–one law''' is a [[theorem]] in [[probability theory]], similar to [[Kolmogorov's zero–one law]] and the [[Borel–Cantelli lemma]], that specifies that a certain type of event will either [[almost surely]] happen or almost surely not happen. It is sometimes known as the '''Hewitt–Savage law for symmetric events'''. It is named after [[Edwin Hewitt]] and [[Leonard Jimmie Savage]].

			==Statement of the Hewitt–Savage zero–one law==
			Let <math>(X_n)_{n = 1}^\infty</math> be a [[sequence]] of [[independent and identically-distributed random variables]] taking values in a set <math>\mathbb{X}</math>. The Hewitt–Savage zero–one law says that any event whose occurrence or non-occurrence is determined by the values of these random variables and whose occurrence or non-occurrence is unchanged by finite [[permutation]]s of the indices, has [[probability]] either 0 or 1 (a "finite" permutation is one that leaves all but finitely many of the indices fixed).

			Somewhat more abstractly, define the ''exchangeable [[sigma algebra]]'' or ''sigma algebra of symmetric events'' <math>\mathcal{E}</math> to be the set of events (depending on the sequence of variables <math>(X_n)_{n = 1}^\infty</math>) which are invariant under [[Wikt:finite\|finite]] [[permutation]]s of the indices in the sequence <math>(X_n)_{n = 1}^\infty</math>. Then <math>A \in \mathcal{E} \implies \mathbb{P} (A) \in \{ 0, 1 \}</math>.

			Since any finite permutation can be written as a product of [[transposition (mathematics)\|transposition]]s, if we wish to check whether or not an event <math>A</math> is symmetric (lies in <math>\mathcal{E}</math>), it is enough to check if its occurrence is unchanged by an arbitrary transposition <math>(i, j)</math>, <math>i, j \in \mathbb{N}</math>.

			==Examples==

			=== Example 1 ===
			Let the sequence <math>(X_n)_{n = 1}^\infty</math> take values in <math>[0, \infty)</math>. Then the event that the series <math>\sum_{n = 1}^\infty X_n</math> converges (to a finite value) is a symmetric event in <math>\mathcal{E}</math>, since its occurrence is unchanged under transpositions (for a finite re-ordering, the convergence or divergence of the series — and, indeed, the numerical value of the sum itself — is independent of the order in which we add up the terms). Thus, the series either converges almost surely or diverges almost surely. If we assume in addition that the common [[expected value]] <math>\mathbb{E}[X_n] > 0</math>, we may conclude that

			:<math>\mathbb{P} \left( \sum_{n = 1}^\infty X_n = + \infty \right) = 1,</math>

			i.e. the series diverges almost surely. This is a particularly simple application of the Hewitt–Savage zero–one law. In many situations, it can be easy to apply the Hewitt–Savage zero–one law to show that some event has probability 0 or 1, but surprisingly hard to determine ''which'' of these two extreme values is the correct one.

			===Example 2===
			Continuing with the previous example, define

			: <math>S_N= \sum_{n = 1}^N X_n,</math>

			which is the position at step ''N'' of a [[random walk]] with the [[iid]] increments ''X''<sub>''n''</sub>. The event { ''S''<sub>''N''</sub> = 0 infinitely often } is invariant under finite permutations. Therefore, the zero–one law is applicable and one infers that the probability of a random walk with real iid increments visiting the origin infinitely often is either one or zero. Visiting the origin infinitely often is a tail event with respect to the sequence (''S''<sub>''N''</sub>), but ''S''<sub>''N''</sub> are not iid and therefore the [[Kolmogorov's zero–one law]] is not directly applicable here. This example is from pages 381 and 382 of the second edition of the probability theory book of [[Albert Shiryaev]].

			== References ==
			* {{cite journal
			\| last1 = Hewitt
			\| first1 = E.
			\| authorlink1 = Edwin Hewitt
			\| last2=Savage
			\| first2=L. J.
			\| authorlink2=Leonard Jimmie Savage
			\| title = Symmetric measures on Cartesian products
			\| journal = Trans. Amer. Math. Soc.
			\| volume = 80
			\| year = 1955
			\| pages = 470–501 }}
			* {{cite book
			\| last = Shiryaev
			\| first = A.
			\| authorlink = Albert Shiryaev
			\| title = Probability Theory
			\| edition = Second
			\| publisher = Springer-Verlag
			\| location = New York
			\| year = 1996 }}

			{{DEFAULTSORT:Hewitt-Savage zero-one law}}
			[[Category:Probability theorems]]
			[[Category:Covering lemmas]]

en>Headbomb: /* References */Various citation cleanup (including MR tyding up), replaced: {{MathSciNet|id=2250060}} → {{MR|2250060}} (2), }} {{MR| → |mr= (2) using AWB

2011-08-21T16:54:49Z

References: Various citation cleanup (including MR tyding up), replaced: {{MathSciNet|id=2250060}} → {{MR|2250060}} (2), }} {{MR| → |mr= (2) using AWB

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Hello buddy. Allow me introduce myself. I am Ron but I don't like when individuals use my full title. The [http://www.Autobytel.com/car-buying-tips/warranty-information/ favorite pastime] for my children and me is taking part in crochet and now [https://Autoclubsouth.Aaa.com/Automotive/auto_warranty.aspx?nvbar=Automotive:ExtendedWarranty I'm attempting] [http://www.chillister.com/blogs/post/27388 auto warranty] to make extended [http://89.1.13.189/index.php?mod=users&action=view&id=63366 car warranty] cash with it. Managing people is how I make cash and it's something I really appreciate. Her spouse and her extended car warranty selected to reside in Delaware but she requirements to move simply extended auto [http://Autos.Jdpower.com/content/buying-tip/xa345A7/pros-and-cons-of-buying-an-extended-warranty-for-a-car.htm warranty] because of her family members.<br><br>my web blog ... [http://xn--h1aecfj6f.xn--p1ai/content/auto-repair-tips-youll-wish-youd-read-sooner ильком.xn--p1ai]

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en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q4669933

en>Headbomb: /* References */Various citation cleanup (including MR tyding up), replaced: {{MathSciNet|id=2250060}} → {{MR|2250060}} (2), }} {{MR| → |mr= (2) using AWB