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en>Bender235 at 11:07, 24 July 2013

2013-07-24T11:07:11Z

← Older revision		Revision as of 13:07, 24 July 2013
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	~~Greetings! I am Marvella and I really feel comfortable~~ when ~~individuals use the full title~~. ~~Hiring has been my profession~~ for ~~some~~ time ~~but I~~'~~ve currently applied for another~~ 1. ~~Years ago we moved to North Dakota~~. ~~What I adore performing is performing ceramics but I haven~~'t ~~made a dime with it~~.<br><br>~~My web site~~: ~~std testing at home~~ (~~[http~~://~~dore~~.~~gia~~.~~ncnu~~.~~edu~~.tw/~~88ipart/node/1326254 reference~~])		In [[probability theory]], the [[central limit theorem]] says that, under certain conditions, the sum of many [[independent identically-distributed random variables]], when scaled appropriately, [[converges in distribution]] to a standard [[normal distribution]]. The '''martingale central limit theorem''' generalizes this result for random variables to [[Martingale (probability theory)\|martingale]]s, which are [[stochastic process]]es where the change in the value of the process from time ''t'' to time ''t'' + 1 has [[expected value\|expectation]] zero, even conditioned on previous outcomes.

			==Statement==
			Here is a simple version of the martingale central limit theorem: Let
			:<math>X_1, X_2, \dots\,</math> -- be a martingale with bounded increments, i.e., suppose

			:<math>\operatorname{E}[X_{t+1} - X_t \vert X_1,\dots, X_t]=0\,,</math>

			and

			:<math>\|X_{t+1} - X_t\| \le k</math>

			[[almost surely]] for some fixed bound ''k'' and all ''t''. Also assume that <math>\|X_1\|\le k</math> almost surely.

			Define

			:<math>\sigma_t^2 = \operatorname{E}[(X_{t+1}-X_t)^2\|X_1, \ldots, X_t],</math>

			and let

			:<math>\tau_\nu = \min\left\{t : \sum_{i=1}^{t} \sigma_i^2 \ge \nu\right\}.</math>

			Then

			:<math>\frac{X_{\tau_\nu}}{\sqrt{\nu}}</math>

			converges in distribution to the normal distribution with mean 0 and variance 1 as <math>\nu \to +\infty \!</math>. More explicitly,

			:<math>\lim_{\nu \to +\infty} \operatorname{P} \left(\frac{X_{\tau_\nu}}{\sqrt{\nu}} < x\right) = \Phi(x)
			= \frac{1}{\sqrt{2\pi}}
			\int_{-\infty}^x
			\exp\left(-\frac{u^2}{2}\right)
			\, du, \quad x\in\mathbb{R}.
			</math>

			==References==
			Many other variants on the martingale central limit theorem can be found in:
			*{{cite book \| first = Peter \| last = Hall \| coauthors = and C. C. Heyde \| year = 1980 \| title = Martingale Limit Theory and Its Application \| publisher = Academic Press \| location = New York \| isbn = 0-12-319350-8}}

			*For the discussion of Theorem 5.4 there, and correct form of Corollary 5.3(ii), see {{cite journal\|
			last=Bradley\| first=Richard\|
			journal=Journal of Theoretical Probability\|
			volume=1\|
			pages=115–119\|
			year=1988\|
			publisher=Springer \| title=On some results of MI Gordin: a clarification of a misunderstanding\|
			doi=10.1007/BF01046930\|
			issue=2}}

			[[Category:Martingale theory]]
			[[Category:Central limit theorem]]

en>Urod at 11:31, 28 April 2012

2012-04-28T11:31:29Z

New page

Greetings! I am Marvella and I really feel comfortable when individuals use the full title. Hiring has been my profession for some time but I've currently applied for another 1. Years ago we moved to North Dakota. What I adore performing is performing ceramics but I haven't made a dime with it.<br><br>My web site: std testing at home ([http://dore.gia.ncnu.edu.tw/88ipart/node/1326254 reference])

Isotonic regression - Revision history

en>SporkBot: Repair duplicate template args

en>Bender235 at 11:07, 24 July 2013

en>Urod at 11:31, 28 April 2012