1. Dynkin systems - Probability

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Tutorial 1: Dynkin systems11. Dynkin systemsDefinition 1ADynkin systemon a set is a subsetDof thepower setP( ), with the following properties:(i) D(ii)A, B D,A B B\A D(iii)An D,An An+1,n 1 + n=1An DDefinition 2A -algebraon a set is a subsetFof the powersetP( )with the following properties:(i) F(ii)A F Ac = \A F(iii)An F,n 1 + n=1An 1: Dynkin systems2Exercise a -algebra on . Show that F,thatifA, B FthenA B Fand alsoA B F. Recall thatB\A=B Acand conclude thatFis also a Dynkin system on .Exercise (Di)i Ibe an arbitrary family of Dynkin systemson , withI = . Show thatD = i IDiis also a Dynkin system on.

Tutorial 1: Dynkin systems 3 Show that D(A) is a Dynkin system on Ω such that A⊆D(A), and that it is the smallest Dynkin system on Ω with such property, (i.e. if D is a Dynkin system on Ω with A⊆D,thenD(A) ⊆D). Definition 3 Let A⊆P(Ω).WecallDynkin system generated by A, the Dynkin system on Ω,denotedD(A), equal to the intersection of all Dynkin systems on Ω, which contain A.

  System, Probability, Dynkin systems, Dynkin