Doob's martingale convergence theorems

Doob's martingale convergence theorems In mathematics – specifically, in the theory of stochastic processes – Doob's martingale convergence theorems are a collection of results on the limits of supermartingales, named after the American mathematician Joseph L. Doob.[1] Informell, the martingale convergence theorem typically refers to the result that any supermartingale satisfying a certain boundedness condition must converge. One may think of supermartingales as the random variable analogues of non-increasing sequences; from this perspective, the martingale convergence theorem is a random variable analogue of the monotone convergence theorem, which states that any bounded monotone sequence converges. There are symmetric results for submartingales, which are analogous to non-decreasing sequences.

Inhalt 1 Statement for discrete-time martingales 1.1 Beweisskizze 1.2 Failure of convergence in mean 2 Statements for the general case 2.1 Doob's first martingale convergence theorem 2.2 Doob's second martingale convergence theorem 3 Doob's upcrossing inequality 4 Anwendungen 4.1 Convergence in Lp 4.2 Lévy's zero–one law 5 Siehe auch 6 References Statement for discrete-time martingales A common formulation of the martingale convergence theorem for discrete-time martingales is the following. Lassen {Anzeigestil X_{1},X_{2},X_{3},Punkte } be a supermartingale. Suppose that the supermartingale is bounded in the sense that {displaystyle sup _{tin mathbf {N} }Name des Bedieners {E} [X_{t}^{-}]0} dann {Anzeigestil Y_{n+1}=Y_{n}pm 1} , Also {Anzeigestil Y} is almost surely zero.

Das bedeutet, dass {Anzeigestil Betreibername {E} [Y]=0} . Jedoch, {Anzeigestil Betreibername {E} [Y_{n}]=1} für jeden {Anzeigestil ngeq 1} , seit {Anzeigestil (Y_{n})_{nin mathbf {N} }} is a random walk which starts at {Anzeigestil 1} and subsequently makes mean-zero moves (alternately, note that {Anzeigestil Betreibername {E} [Y_{n}]= Betreibername {E} [Y_{0}]=1} seit {Anzeigestil (Y_{n})_{nin mathbf {N} }} is a martingale). Deswegen {Anzeigestil (Y_{n})_{nin mathbf {N} }} cannot converge to {Anzeigestil Y} in mean. Darüber hinaus, wenn {Anzeigestil (Y_{n})_{nin mathbb {N} }} were to converge in mean to any random variable {Anzeigestil R} , then some subsequence converges to {Anzeigestil R} fast sicher. So by the above argument {displaystyle R=0} fast sicher, which contradicts convergence in mean.

Statements for the general case In the following, {Anzeigestil (Omega ,F,F_{*},mathbf {P} )} will be a filtered probability space where {Anzeigestil F_{*}=(F_{t})_{tgeq 0}} , und {Anzeigestil N:[0,unendlich )times Omega to mathbf {R} } will be a right-continuous supermartingale with respect to the filtration {Anzeigestil F_{*}} ; mit anderen Worten, für alle {displaystyle 0leq sleq t<+infty } , {displaystyle N_{s}geq operatorname {E} {big [}N_{t}mid F_{s}{big ]}.} Doob's first martingale convergence theorem Doob's first martingale convergence theorem provides a sufficient condition for the random variables {displaystyle N_{t}} to have a limit as {displaystyle tto +infty } in a pointwise sense, i.e. for each {displaystyle omega } in the sample space {displaystyle Omega } individually. For {displaystyle tgeq 0} , let {displaystyle N_{t}^{-}=max(-N_{t},0)} and suppose that {displaystyle sup _{t>0}Name des Bedieners {E} {groß [}N_{t}^{-}{groß ]}<+infty .} Then the pointwise limit {displaystyle N(omega )=lim _{tto +infty }N_{t}(omega )} exists and is finite for {displaystyle mathbf {P} } -almost all {displaystyle omega in Omega } .[3] Doob's second martingale convergence theorem It is important to note that the convergence in Doob's first martingale convergence theorem is pointwise, not uniform, and is unrelated to convergence in mean square, or indeed in any Lp space. In order to obtain convergence in L1 (i.e., convergence in mean), one requires uniform integrability of the random variables {displaystyle N_{t}} . By Chebyshev's inequality, convergence in L1 implies convergence in probability and convergence in distribution. The following are equivalent: {displaystyle (N_{t})_{t>0}} is uniformly integrable, d.h. {Anzeigestil lim _{Cto infty }sup _{t>0}int _{{omega in Omega ,Mitte ,|N_{t}(Omega )|>C}}links|N_{t}(Omega )Rechts|,Mathrm {d} mathbf {P} (Omega )=0;} there exists an integrable random variable {displaystyle Nin L^{1}(Omega ,mathbf {P} ;mathbf {R} )} so dass {Anzeigestil N_{t}to N} wie {displaystyle tto infty } beide {Anzeigestil mathbf {P} } -almost surely and in {Anzeigestil L^{1}(Omega ,mathbf {P} ;mathbf {R} )} , d.h. {Anzeigestil Betreibername {E} links[links|N_{t}-Nright|Rechts]=int _{Omega }links|N_{t}(Omega )-N(Omega )Rechts|,Mathrm {d} mathbf {P} (Omega )zu 0{Text{ wie }}tto +infty .} Doob's upcrossing inequality The following result, called Doob's upcrossing inequality or, sometimes, Doob's upcrossing lemma, is used in proving Doob's martingale convergence theorems.[3] EIN "gambling" argument shows that for uniformly bounded supermartingales, the number of upcrossings is bounded; the upcrossing lemma generalizes this argument to supermartingales with bounded expectation of their negative parts.

Lassen {Anzeigestil N} be a natural number. Lassen {Anzeigestil (X_{n})_{nin mathbf {N} }} be a supermartingale with respect to a filtration {Anzeigestil ({mathematisch {F}}_{n})_{nin mathbf {N} }} . Lassen {Anzeigestil a} , {Anzeigestil b} be two real numbers with {Anzeigestil aThese are called upcrossings with respect to interval {Anzeigestil [a,b]} . Dann {Anzeigestil (b-a)Name des Bedieners {E} [U_{n}]leq operatorname {E} [(X_{n}-a)^{-}].Quad } wo {Anzeigestil X^{-}} is the negative part of {Anzeigestil X} , definiert von {textstyle X^{-}=-min(X,0)} .[4][5] Applications Convergence in Lp Let {Anzeigestil M:[0,unendlich )times Omega to mathbf {R} } be a continuous martingale such that {displaystyle sup _{t>0}Name des Bedieners {E} {groß [}{groß |}M_{t}{groß |}^{p}{groß ]}<+infty } for some {displaystyle p>1} . Then there exists a random variable {displaystyle Min L^{p}(Omega ,mathbf {P} ;mathbf {R} )} so dass {Anzeigestil M_{t}to M} wie {displaystyle tto +infty } beide {Anzeigestil mathbf {P} } -almost surely and in {Anzeigestil L^{p}(Omega ,mathbf {P} ;mathbf {R} )} .

The statement for discrete-time martingales is essentially identical, with the obvious difference that the continuity assumption is no longer necessary.

Lévy's zero–one law Doob's martingale convergence theorems imply that conditional expectations also have a convergence property.

Lassen {Anzeigestil (Omega ,F,mathbf {P} )} be a probability space and let {Anzeigestil X} be a random variable in {Anzeigestil L^{1}} . Lassen {Anzeigestil F_{*}=(F_{k})_{kin mathbf {N} }} be any filtration of {Anzeigestil F} , und definieren {Anzeigestil F_{unendlich }} to be the minimal σ-algebra generated by {Anzeigestil (F_{k})_{kin mathbf {N} }} . Dann {Anzeigestil Betreibername {E} {groß [}Xmid F_{k}{groß ]}to operatorname {E} {groß [}Xmid F_{unendlich }{groß ]}{Text{ wie }}kto infty } beide {Anzeigestil mathbf {P} } -almost surely and in {Anzeigestil L^{1}} .

This result is usually called Lévy's zero–one law or Levy's upwards theorem. The reason for the name is that if {Anzeigestil A} is an event in {Anzeigestil F_{unendlich }} , then the theorem says that {Anzeigestil mathbf {P} [Amid F_{k}]to mathbf {1} _{EIN}} fast sicher, d.h., the limit of the probabilities is 0 oder 1. In plain language, if we are learning gradually all the information that determines the outcome of an event, then we will become gradually certain what the outcome will be. This sounds almost like a tautology, but the result is still non-trivial. Zum Beispiel, it easily implies Kolmogorov's zero–one law, since it says that for any tail event A, we must have {Anzeigestil mathbf {P} [EIN]=mathbf {1} _{EIN}} fast sicher, somit {Anzeigestil mathbf {P} [EIN]in {0,1}} .

Similarly we have the Levy's downwards theorem : Lassen {Anzeigestil (Omega ,F,mathbf {P} )} be a probability space and let {Anzeigestil X} be a random variable in {Anzeigestil L^{1}} . Lassen {Anzeigestil (F_{k})_{kin mathbf {N} }} be any decreasing sequence of sub-sigma algebras of {Anzeigestil F} , und definieren {Anzeigestil F_{unendlich }} to be the intersection. Dann {Anzeigestil Betreibername {E} {groß [}Xmid F_{k}{groß ]}to operatorname {E} {groß [}Xmid F_{unendlich }{groß ]}{Text{ wie }}kto infty } beide {Anzeigestil mathbf {P} } -almost surely and in {Anzeigestil L^{1}} .

See also Backwards martingale convergence theorem[6] Dieser Artikel enthält eine Liste allgemeiner Referenzen, aber es fehlen genügend entsprechende Inline-Zitate. Bitte helfen Sie mit, diesen Artikel zu verbessern, indem Sie genauere Zitate einfügen. (Januar 2012) (Erfahren Sie, wie und wann Sie diese Vorlagennachricht entfernen können) References ^ Doob, J. L. (1953). Stochastic Processes. New York: Wiley. ^ Durrett, Rick (1996). Wahrscheinlichkeit: theory and examples (Zweite Aufl.). Duxbury Press. ISBN 978-0-534-24318-0.; Durrett, Rick (2010). 4te Auflage. ISBN 9781139491136. ^ Nach oben springen: ein b "Martingale Convergence Theorem" (Pdf). Massachusetts Institute of Tecnnology, 6.265/15.070J Lecture 11-Additional Material, Advanced Stochastic Processes, Herbst 2013, 10/9/2013. ^ Bobrowski, Adam (2005). Functional Analysis for Probability and Stochastic Processes: Eine Einleitung. Cambridge University Press. pp. 113–114. ISBN 9781139443883. ^ Gushchin, EIN. EIN. (2014). "On pathwise counterparts of Doob's maximal inequalities". Proceedings of the Steklov Institute of Mathematics. 287 (287): 118–121. arXiv:1410.8264. doi:10.1134/S0081543814080070. S2CID 119150374. ^ Doob, Joseph L. (1994). Measure theory. Abschlusstexte in Mathematik, Vol. 143. Springer. p. 197. ISBN 9781461208778. Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN 3-540-04758-1. (See Appendix C) Kategorien: Probability theoremsMartingale theory