For the martingale betting strategy martingale betting system Image HittingTimes1.png thumb 340px Stopped process Brownian motion Stopped Brownian motion is an example of a martingale. It can be used to model an even coin toss betting game with the possibility of bankruptcy. In probabilitytheory , a martingale ... process Brownian motion Stopped Brownian motion , which is a martingale process, can be used to model the trajectory of such games. The concept of martingale in probabilitytheory was introduced by Paul ... to Martingaleprobabilitytheory. cite book author link David Williams mathematician first David ... of being a martingale involves both the filtration and the probability measure with respect to which ... Moivre de Moivre s martingale Now suppose an unfair or biased coin, with probability p of heads and probability ... variables. This sequence is a martingale under the unified neutral theory of biodiversity . If N ... a relationship between martingaletheory and potential theory , which is the study of harmonic function ... processes Category Martingaletheory Category Game theory de Martingal es Martingala fr Martingale .... In particular, a martingale is a sequence of random variable s i.e., a stochastic process for which, at a particular time in the realization probability realized sequence, the Expected value ... of all prior realization probability observed value s at a current time. To contrast, in a process that is not a martingale, it may still be the case that the expected value of the process at one .... History Originally, martingale betting system martingale referred to a class of betting strategy ... of the Word Martingale last1 Mansuy first1 Roger month June year 2009 volume 5 number 1 journal Electronic Journal for History of Probability and Statistics accessdate 10 22 2011 ref The simplest of these strategies ... stake. As the gambler s wealth and available time jointly approach infinity, his probability of eventually flipping heads approaches 1, which makes the martingale betting strategy seem like a almost surely ... more details
Martingale can refer to Martingaleprobabilitytheory , a stochastic process in which the conditional expectation of the next value, given the current and preceding values, is the current value Martingale tack for horses Martingale collar for dogs and other animals Martingale betting system Martingale rigging In the sport of fencing , a martingale is a strap attached to the sword handle to prevent a sword from being dropped if disarmed. In the theatrical lighting industry, martingale is an obsolete term for a twofer , or occasionally a threefer. disambig eo Martingalo fr Martingale homonymie it Martingala disambigua sv Martingal zh ... more details
Martingale pricing is a pricing approach based on the notions of Martingaleprobabilitytheorymartingale and risk neutral measure risk neutrality . The martingale pricing approach is a cornerstone of modern quantitative finance and can be applied to a variety of derivative finance derivatives contracts, e.g. option finance options , Futures contract futures , interest rate derivative s, credit derivatives , etc. In contrast to the Partial differential equation PDE approach to pricing, martingale pricing formulae are in the form of expectations which can be efficiently solved numerically using a Monte Carlo method Monte Carlo approach. As such, Martingale pricing is preferred when valuing highly dimensional contracts such as a basket of options. On the other hand, valuing American option American style contracts is troublesome and requires discretizing the problem making it like a Bermudan option and and only in 2001 Francis Longstaff F. A. Longstaff and Eduardo Schwartz E. S. Schwartz developed a practical Monte Carlo method for pricing American options. ref cite journal last1 Longstaff first1 F.A. first2 E.S. last2 Schwartz url http repositories.cdlib.org anderson fin 1 01 accessdate October 8, 2011 title Valuing American options by simulation a simple least squares approach journal Review of Financial Studies volume 14 year 2001 pages 113 148 ref See also Martingaleprobabilitytheory References Reflist DEFAULTSORT Martingale Pricing Category Finance theories Category Mathematical finance Category Pricing ... more details
Refimprove date September 2009 ProbabilityTopics Probabilitytheory is the branch of mathematics concerned ... www.britannica.com ebc article 9375936 title Probabilitytheory, Encyclopaedia Britannica publisher Britannica.com date accessdate 2012 02 12 ref The central objects of probabilitytheory are random variable s, stochastic process es, and event probabilitytheory event s mathematical abstractions ... foundation for statistics , probabilitytheory is essential to many human activities that involve quantitative analysis of large sets of data. Methods of probabilitytheory also apply to descriptions ... scales, described in quantum mechanics . History The mathematical theory of probability has ... Introduction ref Initially, probabilitytheory mainly considered discrete events, and its methods ... probabilitytheory, on foundations laid by Andrey Nikolaevich Kolmogorov . Kolmogorov combined the notion ... axioms axiom system for probabilitytheory in 1933. Fairly quickly this became the mostly undisputed axiom system axiomatic basis for modern probabilitytheory but alternatives exist, in particular ... introductions to probabilitytheory treat discrete probability distributions and continuous probability distributions separately. The more mathematically advanced measure theory based treatment of probability ... Discrete probabilitytheory deals with events that occur in countable sample spaces. Examples ... to 1. An Event probabilitytheory event is defined as any subset math E , math of the sample space ... how probability mass functions are obtained instead it builds a theory that assumes their existence ... continuous sample spaces. Measure theoretic probabilitytheory The raison d tre of the measure theoretic .... The modern approach to probabilitytheory solves these problems using measure theory to define the probability ... have gained special importance in probabilitytheory. Some fundamental discrete distributions ... of random variables In probabilitytheory, there are several notions of convergence for random variable ... more details
context date February 2012 In the mathematical theory of probabilitytheoryprobability , a sigma martingale is a semimartingale with an integral representation. Sigma martingales were introduced by C.S. Chou and M. Emery in 1977 and 1978. ref name fundamental theorem Not every local martingale is a sigma martingale. In financial mathematics , sigma martingales appear in the fundamental theorem of asset pricing as an equivalent condition to no free lunch with vanishing risk a no arbitrage condition . ref cite journal title What is... a Free Lunch? first1 Freddy last1 Delbaen first2 Walter last2 Schachermayer journal Notices of the AMS volume 51 number 5 pages 526 528 url http www.ams.org notices 200405 what is.pdf format pdf accessdate October 14, 2011 ref Mathematical definition An math mathbb R d math valued stochastic process math X X t t 0 T math is a sigma martingale if it is a semimartingale and there exists an math mathbb R d math valued martingale M and an M Ito integral integrable predictable process math phi math with values in math mathbb R math such that math X phi cdot M. , math ref name fundamental theorem cite journal title The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes author1 F. Delbaen author2 W. Schachermayer journal Mathematische Annalen year 1998 volume 312 pages 215 250 url http www.mat.univie.ac.at schachermayer pubs preprnts prpr0084.pdf format pdf accessdate October 14, 2011 doi 10.1007 s002080050220 ref References Reflist Category Stochastic processes Category Martingaletheoryprobability stub ... more details
Martingaletheoryprobability stub ru uk vi Martingale Doob ... Bernstein inequalities probabilitytheory nofootnotes date March 2011 References cite journal ...refimprove date March 2011 context date March 2011 A Doob martingale also known as a Levy martingale is a mathematical construction of a stochastic process which approximates a given random variable and has the Martingaleprobabilitytheorymartingale property with respect to the given filtration mathematics filtration . It may be thought of as the evolving sequence of best approximations to the random variable based on information accumulated up to a certain time. When analyzing sums, random walk s, or other additive functions of Statistical independence independent random variables , one can often apply the central limit theorem , law of large numbers , Chernoff s inequality , Chebyshev s inequality or similar tools. When analyzing similar objects where the differences are not independent, the main tools are Martingaleprobabilitytheorymartingale s and Azuma s inequality . Definition A Doob martingale named after Joseph Leo Doob J. L. Doob cn date March 2011 is a generic construction that is always a martingale. Specifically, consider any set of random variables math vec X X 1, X 2, ..., X n math taking values in a set math A math for which we are interested in the function math f A n to Bbb R math and define math B i E X i 1 ,X i 2 ,...,X n f vec X X 1 ,X 2 ,...X i math where the above expectation is itself a random quantity since the expectation is only taken over math X i 1 ,X i 2 ,...,X n , math and math X 1 ,X 2 ,...X i math are treated as random variables. It is possible to show that math B i math is always a martingale regardless of the properties of math X i math ... that with high probability math f vec X math is concentrated around its expected value math E f ... s inequality to a Doob martingale is called McDiarmid s inequality. cn date March 2011 Suppose math ... more details
2003 isbn 3 540 04758 1 DEFAULTSORT Local Martingale Category Martingaletheory de Lokales Martingal ...In mathematics , a local martingale is a type of stochastic process , satisfying the Stopping time Localization localized version of the Martingaleprobabilitytheorymartingale property. Every martingale is a local martingale every bounded local martingale is a martingale however, in general a local martingale is not a martingale, because its expectation can be distorted by large values of small probability. In particular, a It diffusion driftless diffusion process is a local martingale, but not necessarily a martingale. Local martingales are essential in stochastic analysis , see It calculus ... P be a probability space let F sub sub F sub t sub t 0 ... sub local martingale if there exists a sequence of F sub sub stopping rule stopping times sub k sub ... k math is an F sub &lowast sub martingale for every k . Examples Example 1 Let W sub t sub be the Wiener ... sub is a martingale its expectation is 0 at all times, nevertheless its limit as t &rarr ... le t infty. end cases math This process is not a martingale. However, it is a local martingale. A localizing ..., for all k that exceed the maximal value of the process X . The process stopped at sub k sub is a martingale. ref group details For the times before 1 it is a martingale since a stopped Brownian motion ... mathbb E f W 1 infty. math Then the following process is a martingale math displaystyle X t mathbb ... le t 1, 0 & text for 1 le t infty. end cases math This process is not a martingale. However, it is a local martingale. A localizing sequence may be chosen as math tau k min t Y t k . math Example ... since math Z t math does not hit 1, almost surely , and is a local martingale, since the function ... for r 1 but to 0 for r 1 . Martingales via local martingales Let math M t math be a local martingale. In order to prove that it is a martingale it is sufficient to prove that math M t tau k to M t math ... more details
Unreferenced date December 2009 Image Color BrownPinkDots MED.jpg thumb Martingale Collar with Chain Loop martingale collars also come with a fabric flat tab or loop instead of a chain, and option buckles on both styles. A martingale is a type of dog collar that provides more control over the animal without the choking effect of a slip collar . It is similar in concept to a horse s martingale tack martingale . Martingale Dog collars are also known as Greyhound , Whippet or humane choke collars. The Martingale dog collar was designed for Sighthound s because their necks are larger than their heads and they can often slip out of buckle collars. These collars have gained popularity among other breed owners in the recent past with many trainers now recommending them instead of choke chains or buckle collars. Martingale Dog Collars are fitted to the exact size of the dog s neck when the collar is pulled closed. Properly fitted, the collar will be comfortably loose when not in use. When the Martingale control loop is pulled, the two slides hardware on the small control loop of the Martingale Collar should pull close together but the two slides should never touch. DEFAULTSORT Martingale Collar Category Dog equipment fr Enr nement La martingale ... more details
File Stad Amsterdam Galleonsfigur.jpg thumb Martingale and Dolphin striker on tall ship Stad Amsterdam A martingale is a fore and aft stay lying directly beneath the bowsprit strengthening it and, if extended from the sprit a jibboom , against upward force created by the head Stays nautical stays . The martingale is part of a vessel s standing rigging . Often a dolphin striker is used between the martingale and bowsprit to provide additional tension to the former. See also boom vang Unreferenced date September 2008 Sail Types Category Sailing rigs and rigging Category Sailboat components ... more details
Other uses Martingale disambiguation Martingale A martingale is any of several designs of horse tack ... of martingale, the standing and the running, are used to control the horse s head height, and to prevent ... livestock poll or upper neck. When a horse s head gets above a desired height, the martingale places ... martingale File Norfolk Hunt Horse Show.jpg right thumb The standing martingale. File Barrel Racing Szmurlo.jpg right thumb The tiedown The standing martingale, also known as a tiedown or a head ... for English riding, it should be possible to push the martingale strap up to touch the horse s equine anatomy throatlatch . A variation of the standing martingale, called a tiedown, is seen almost exclusively in the western riding disciplines. A tiedown is adjusted much shorter than a standing martingale ... harsher than the English style standing martingale. It is properly adjusted when it puts no pressure .... The standing martingale is competition legal for show hunter and hunt seat equitation riders over fences ... in any other western style horse show competition. Safety and risks The standing martingale is more restrictive than the running martingale because it cannot be loosened in an emergency. A horse that trips in a standing martingale could potentially fall more easily because its range of motion is restricted. If a horse falls wearing an incorrectly fitted standing martingale, the animal cannot extend ... of the nose, the martingale strap is never attached to a noseband drop noseband . Because ... 8 or grackle noseband. A standing martingale can be attached to the cavesson the upper, heavier strap of a noseband flash noseband , but not to the lower, flash or drop strap. Any martingale may ... with a gag bit , a standing martingale can trap the head of the horse, simultaneously asking the horse .... Overuse or misuse of a martingale or tiedown, particularly as a means to prevent a horse from ... of accidents If a horse is sufficiently trapped by a combination of a too short martingale and too ... more details
Infobox Film name Fifi Martingale image image size caption director Jacques Rozier producer writer narrator starring Jean Lefebvre br Lili Vonderfeld music cinematography editing distributor released 2001 runtime 127 minutes country FilmFrance language French budget preceded by followed by Fifi Martingale is a 2001 French film directed by Jacques Rozier . The film is about a theater company attempting to put on a new play. It was Jean Lefebvre s last film. Cast Jean Lefebvre as Gaston Manzanar s Lili Vonderfeld as Fifi Mike Marshall actor Mike Marshall as the Author Jacques Petitjean as the Director Yves Afonso as Yves Lempereur Fran ois Chattot as P re Popelkov Alexandra Stewart as the Ambassador Jacques Fran ois as the Ambassador Roger Trapp as The consul of Moldova External links imdb title 0143088 Category French films Category 2001 films 2000s France film stub ... more details
In probabilitytheory , a martingale difference sequence MDS is related to the concept of the martingaleprobabilitytheorymartingale . A stochastic process stochastic series Y is an MDS if its expected value expectation with respect to past values of another stochastic series X is zero. Formally math E Y i 1 X i ,X i 1 , dots 0 quad text for all i. math If math Z math is a martingale, for example, then math Y i Z i Z i 1 math will be an MDS hence the name. References James Douglas Hamilton 1994 , Time Series Analysis , Princeton University Press. ISBN 0 691 04289 6 Category Stochastic processes Category Martingaletheoryprobability stub it Differenza di martingala ... more details
F R br Doob martingale F R br Independence probabilitytheory Independence F BR br Littlewood Offord problem F R br L vy flight F R U C br MartingaleprobabilitytheoryMartingale FU R br Martingale difference sequence F R br Mid Maximum likelihood ... U R br MartingaleprobabilitytheoryMartingale FU R br Stationary process SU R br Stochastic ...ProbabilityTopicsTOC This page lists articles related to probabilitytheory . In particular, it lists many articles corresponding to specific probability distributions . Such articles are marked here by a code .... Core probability selected topics Probabilitytheory Basic notions bsc Top Random variable br Continuous probability distribution 1 C br Cumulative distribution function 1 DCR br Discrete probability distribution 1 D br Independent and identically distributed random variables ... Zygmund inequality inq br Method of moments probabilitytheory Method of moments ... F R br Bernstein inequalities probabilitytheory Bernstein inequalities F R ... probabilitytheory Method of moments mnt L R br Slutsky s theorem anl br Weak convergence ... function probabilitytheory Characteristic function lmt 1F DCR br Contiguity Probability ... 1 BDC br Mid Event probabilitytheory Event 1 B br Indecomposable distribution 1 BDCR ... 2 R br Total variation Total variation distance in probabilitytheory 2 R br Bottom General ... F B br Inclusion exclusion principle F B br Independence probabilitytheory Independence ... FU DG br Bottom Continuous F C Top Anderson s theorem Application to probabilitytheory ... LS R br Doob s martingale convergence theorems SU R br Ergodic theory S R br ... Dutch book br Elementary event br Mid Normalizing constant br Possibility theory br Probability axioms ... probability geo Top Boolean model probabilitytheory Boolean model br Buffon s needle br Geometric ... more details
In probabilitytheory , 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 Martingaleprobabilitytheorymartingale 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 Martingaletheory Category ... more details
In probabilitytheory , the martingale representation theorem states that a random variable which is measurable with respect to the Filtration mathematics Measure theory filtration generated by a Brownian motion can be written in terms of an It integral with respect to this Brownian motion. The theorem only asserts the existence of the representation and does not help to find it explicitly it is possible in many cases to determine the form of the representation using Malliavin calculus . Similar theorems also exist for Martingaleprobabilitytheory martingales on filtrations induced by jump processes, for example, by Markov chain s. Statement of the theorem Let math B t math be a Brownian motion on a standard filtered probability space math Omega, mathcal F , mathcal F t, P math and let math mathcal G t math be the augmentation of the filtration generated by math B math . If X is a square integrable random variable measurable with respect to math mathcal G infty math , then there exists a predictable process C which is adapted process adapted with respect to math mathcal G t math , such that math X E X int 0 infty C s ,dB s. math Consequently math E X mathcal G t E X int 0 t C s , d B s. math Application in finance The martingale representation theorem can be used to establish the existence of a hedging strategy. Suppose that math left M t right 0 le t infty math is a Q martingale process, whose volatility math sigma t math is always non zero. Then, if math left N t right 0 le t infty math is any other Q martingale, there exists an F previsible process math phi math , unique up to sets of measure 0, such that math int 0 T phi t 2 sigma t 2 , dt infty math with probability one, and N can be written as math N t N 0 int 0 t phi s , d M s. math The replicating strategy is defined to be hold math phi t math units of the stock at the time t , and hold math psi t C t phi t Z ... 226 Category Martingaletheory Category Probability theorems ... more details
In probabilitytheory, an experiment is any procedure that can be infinitely repeated and has a well defined set of outcomes known as the sample space . More formally, an experiment is specified by a tuple S , F , P where S is a Sample space , F is a Borel set specifying a set of events and P is a probability measure which allows the calculation of probabilities for all the events. ref Ludemann, L C 2003 . Random Processes Filtering, Estimation, and Detection , p. 1. Wiley. ISBN 9788126527236. ref An experiment is said to be random if it has more than one possible outcome, and deterministic if it has only one. A random experiment that has two mutually exclusive outcomes is known as a Bernoulli trial . probability stub References references Category Probabilitytheory uk ... more details
Dablink This article is about the method of moments in probabilitytheory . See method of moments for other techniques bearing the same name. In probabilitytheory , the method of moments is a way of proving convergence in distribution by proving convergence of a sequence of moment mathematics moment sequences. ref cite book last Prokhorov first A.V. chapter Moments, method of in probabilitytheory title Encyclopaedia of Mathematics online isbn 1402006098 url http eom.springer.de m m064610.htm mr 1375697 editor M. Hazewinkel ref Suppose X is a random variable and that all of the moments math operatorname E X k , math exist. Further suppose the probability distribution of X is completely determined by its moments, i.e., there is no other probability distribution with the same sequence of moments cf. the problem of moments . If math lim n to infty operatorname E X n k operatorname E X k , math for all values of k , then the sequence X sub n sub converges to X in distribution. The method of moments was introduced by Pafnuty Chebyshev for proving the central limit theorem Chebyshev cited earlier contributions by Ir n e Jules Bienaym ref cite book mr 2743162 last Fischer first H. title A history of the central limit theorem. From classical to modern probabilitytheory. series Sources and Studies in the History of Mathematics and Physical Sciences publisher Springer location New York year 2011 isbn 978 0 387 87856 0 chapter 4. Chebyshev s and Markov s Contributions. ref . More recently, it has been applied by Eugene Wigner to prove Wigner s semicircle law , and has since found numerous applications in the random matrix theorytheory of random matrices . ref cite book last Anderson first G.W. last2 Guionnet first2 A. last3 Zeitouni first3 O. title An introduction to random matrices. year 2010 publisher Cambridge University Press location Cambridge isbn 978 0 521 19452 5 chapter 2.1 ref Notes Reflist DEFAULTSORT Method Of Moments ProbabilityTheory Category Probabilitytheory ... more details
Infobox journal title ProbabilityTheory and Related Fields cover abbreviation Probab. Theory Related Fields discipline Probability editor nowrap 1 G rard Ben Arous , nowrap 1 Amir Dembo publisher Springer Science Business Media Springer frequency Monthly history 1962 present impact 1.59 impact year 2010 url http www.springer.com mathematics probability journal 440 ISSN 0178 8051 eISSN 1432 2064 CODEN PTRFEU LCCN 86650503 OCLC link1 http www.springerlink.com content 1432 2064 link1 name Online access ProbabilityTheory and Related Fields is a peer review peer reviewed mathematics journal published by Springer Science Business Media Springer . Established in 1962, it was originally named Zeitschrift f r Wahrscheinlichkeitstheorie und verwandte Gebiete , with the English replacing the German starting from volume 71 1986 . The journal publishes articles on probability . The journal is indexed by Mathematical Reviews and Zentralblatt MATH . Its 2009 Mathematical Citation Quotient MCQ was 1.19, and its 2009 impact factor was 1.373. External links Official 1 http www.springer.com mathematics probability journal 440 Category Probability journals Category Publications established in 1962 Category English language journals Category Springer academic journals Category Monthly journals math journal stub ... more details
Infobox journal title Theory of Probability and its Applications cover abbreviation Theory Probab. Appl. discipline Probability , statistics editor publisher Society for Industrial and Applied Mathematics SIAM frequency Quarterly history 1956 present impact 0.827 impact year 2009 url http epubs.siam.org tvp ISSN 0040 585X eISSN 1095 7219 CODEN TPRBAU LCCN 61047747 OCLC link1 http epubs.siam.org tvp resource 1 tprbau link1 name Online access Theory of Probability and its Applications is a peer review peer reviewed mathematics journal published quarterly by Society for Industrial and Applied Mathematics SIAM . Established in 1956, the journal is a translation of the Russian journal Teoriya Veroyatnostei i ee Primeneniya . The journal is indexed by Mathematical Reviews and Zentralblatt MATH . Its 2009 Mathematical Citation Quotient MCQ was 0.12, and its 2009 impact factor was 0.827. External links Official 1 http epubs.siam.org tvp Category Publications established in 1956 Category English language journals Category SIAM academic journals Category Quarterly journals Category Probability journals math journal stub ... more details
Other uses E net disambiguation net unreferenced date October 2010 lowercase title net An math varepsilon math net is any of several related concepts in mathematics , and has a particular meaning in probability theory where it is used in desription of the approximation of one probability distribution by another. Theory Let math P math be a probability distribution over some set math X math . An math varepsilon math net for a class math H subseteq 2 X math of subsets of math X math is any subset math S subseteq X math such that for any math h in H math math P h ge varepsilon quad Longrightarrow quad S cap h neq varnothing. math Intuitively math S math approximates the probability distribution. A stronger notion is math varepsilon math approximation. An math varepsilon math approximation for class math H math is a subset math S subseteq X math such that for any math h in H math it holds math left P h frac S cap h S right varepsilon . math References Category Probability theory probability stub ... more details
refimprove date February 2012 one source date February 2012 In probabilitytheory , an event is a Set mathematics set of outcomes a subset of the sample space to which a probability is assigned. ref Leon Garcia, Alberto. Probability, Statistics and Random Processes for Electrical Engineering. Upper Saddle River, NJ Pearson, 2008. Print. ref Typically, when the sample space is finite, any subset of the sample space is an event i . e . all elements of the power set of the sample space are defined as events ... to a more limited family of subsets. For the standard tools of probabilitytheory, such as joint probability joint and conditional probability conditional probabilities , to work, it is necessary ... Reflist DEFAULTSORT Event ProbabilityTheory Category Probabilitytheory bn de ... infinite , most notably when the outcome is a real numbers real number . So, when defining a probability ... being events see Events in probability spaces Events in probability spaces , below . A simple example ... event , with probability zero and the sample space itself a certain event, with probability .... B is the sample space and A is an event. br By the ratio of their areas, the probability of A is approximately ... space is equally likely, the probability of an event A is math mathrm P A frac A Omega , math this rule can readily be applied to each of the example events above. Events in probability spaces Section ... standard probability distributions , such as the normal distribution , the sample space is the set of real ... measurable sets proves more useful in practice. In the general measure theory measure theoretic description of probability space s, an event may be defined as an element of a selected sigma algebra ... that is not an element of the algebra is not an event, and does not have a probability. With a reasonable specification of the probability space, however, all events of interest are elements of the ... for a probability , such as math P u X leq v F v F u ,. math The set mathematics set u X v ... more details
In probabilitytheory and statistics , smoothness of a density function is a measure which determines how many times the density function can be differentiated, or equivalently the limiting behavior of distribution s Characteristic function probabilitytheory characteristic function . Formally, we call the distribution of a random variable X ordinary smooth of order ref name fan91 cite journal last Fan first Jianqing year 1991 title On the optimal rates of convergence for nonparametric deconvolution problems journal The Annals of Statistics volume 19 issue 3 pages 1257 1272 jstor 2241949 doi 10.1214 aos 1176348248 ref if its Characteristic function probabilitytheory characteristic function satisfies math d 0 t beta leq varphi X t leq d 1 t beta quad text as t to infty math for some positive constants d sub 0 sub , d sub 1 sub , . The examples of such distributions are Gamma distribution gamma , Exponential distribution exponential , Uniform distribution continuous uniform , etc. The distribution is called supersmooth of order ref name fan91 if its characteristic function satisfies math d 0 t beta 0 exp big t beta gamma big leq varphi X t leq d 1 t beta 1 exp big t beta gamma big quad text as t to infty math for some positive constants d sub 0 sub , d sub 1 sub , , and constants sub 0 sub , sub 1 sub . Such supersmooth distributions have derivatives of all orders. Examples normal distribution normal , Cauchy distribution Cauchy , mixture normal. References reflist cite book last Lighthill first M. J. year 1962 title Introduction to Fourier analysis and generalized functions publisher London Cambridge University Press Category Theory of probability distributions probability stub ... more details
Image Boolean model.svg right thumb Realization of Boolean model with random radii discs. The Boolean model for a random subset of the plane or higher dimensions, analogously is one of the simplest and most tractable models in stochastic geometry . Take a Poisson process Poisson point process of rate math lambda math in the plane and make each point be the center of a random set the resulting union of overlapping sets is a realization of the Boolean model math mathcal B math . More precisely, the parameters are math lambda math and a probability distribution on compact sets for each point math xi math of the Poisson point process we pick a set math C xi math from the distribution, and then define math mathcal B math as the union math cup xi xi C xi math of translated sets. To illustrate tractability with one simple formula, the mean density of math mathcal B math equals math 1 exp lambda operatorname E Gamma math where math Gamma math denotes the area of math C xi math . The classical theory of stochastic geometry develops many further formulas &ndash see ref cite book author Stoyan, D., Kendall, W.S. and Mecke, J. title Stochastic geometry and its applications year 1987 publisher Wiley ref ref cite book author Schneider, R. and Weil, W. title Stochastic and Integral Geometry year 2008 publisher Springer ref . As related topics, the case of constant sized discs is the basic model of continuum percolation ref cite book author Meester, R. and Roy, R. title Continuum Percolation year 2008 publisher Cambridge University Press ref and the low density Boolean models serve as a first order approximations in the study of extremes in many models ref cite book last Aldous, D. title Probability Approximations via the Poisson Clumping Heuristic year 1988 publisher Springer ref . References references DEFAULTSORT Boolean Model Probability Theory Category Probability theory ... more details
In probabilitytheory , two sequences of probability measure s are said to be contiguous if asymptotically they share the same support measure theory support . Thus the notion of contiguity extends the concept of absolute continuity to the sequences of measures. The concept was originally introduced by harvtxt Le Cam 1960 as part of his contribution to the development of abstract general asymptotic theory in mathematical statistics . Le Cam was instrumental during the period in the development of abstract general asymptotic theory in mathematical statistics. He is best known for the general concepts of local asymptotic normality and contiguity. ??? ref Wolfowitz J. 1974 Review of the book Contiguity of Probability Measures Some Applications in Statistics. by George G. Roussas , Journal of the American Statistical Association , 69, 278&ndash 279 http www.jstor.org pss 2285551 jstor ref Definition Let math style height 1.2em position relative top .2em Omega n, mathcal F n math be a sequence of measurable space s, each equipped with two measures P sub n sub and Q sub n sub . We say that Q ... 1&SRETRY 0 Contiguity of Probability Measures , David J. Scott, La Trobe University http www.jstor.org pss 2242899 On the Concept of Contiguity , Hall, Loynes Category Probabilitytheory ... A 0 . That is, Q is absolutely continuous with respect to P if the support measure theory support of Q ... 200506 fmse course info werker updated nov14.pdf ref See also Contiguity Probability space Notes Reflist References refbegin cite book author H jek, J. coauthor id k, Z. title Theory of rank ... ref CITEREFLe Cam1960 SpringerEOM last Roussas first George G. title Contiguity of probability measures ..., George G. 1972 , Contiguity of Probability Measures Some Applications in Statistics , CUP, ISBN 9780521090957. Scott, D.J. 1982 Contiguity of Probability Measures, Australian & New Zealand Journal of Statistics ... conditions for contiguity and entire asymptotic separation of probability measures R Sh Liptser ... more details
Category Martingaletheory de Doobsche Maximalungleichung vi B t ng th c Doob ...In mathematics , Doob s martingale inequality is a result in the study of stochastic processes . It gives a bound on the probability that a stochastic process exceeds any given value over a given interval of time. As the name suggests, the result is usually given in the case that the process is a non negative Martingaleprobabilitytheorymartingale , but the result is also valid for non negative submartingales. The inequality is due to the USA American mathematician Joseph Leo Doob . Statement of the inequality Let X be a submartingale taking non negative real values, either in discrete or continuous time. That is, for all times s and t with s < t , math mathbf E big X t big mathcal F s big geq X s . math For a continuous time submartingale, assume further that the process is c dl g . Then, for any constant C > 0 and p &ge 1, math mathbf P left sup 0 leq t leq T X t geq C right leq frac mathbf E big X T p big C p . math In the above, as is conventional, P denotes the probability measure on the sample space &Omega of the stochastic process math X 0, T times Omega to 0, infty math and E denotes the expected value with respect to the probability measure P , i.e. the integral math mathbf E big X T big int Omega X T omega , mathrm d mathbf P omega ... of such &sigma algebras forms a filtration abstract algebra filtration of the probability space. Further inequalities There are further sub martingale inequalities also due to Doob. With the same assumptions ... M sub n sub X sub 1 sub ... X sub n sub is a martingale. Note that Jensen s inequality implies that math M n math is a nonnegative submartingale if math M n math is a martingale. Hence, taking p 2 in Doob s martingale inequality, math mathbf P left max ... location Berlin year 1999 isbn 3 540 64325 7 Theorem II.1.7 springer id M m062570 title Martingale author ... more details
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