Στα επόμενα θεωρούμε ότι όλα συμβαίνουν σε ένα χώρο πιθανότητας ( Ω,,P) Modes of convergence: Οι τρόποι σύγκλισης μιας ακολουθίας τ.μ.
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- Κλειώ Μπλέτσας
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1 Στα πόμνα θωρούμ ότι όλα συμβαίνουν σ ένα χώρο πιθανότητας ( Ω,,). Modes of covergece: Οι τρόποι σύγκλισης μιας ακολουθίας τ.μ. { } ίναι οι ξής: σ μια τ.μ.. Ισχυρή σύγκλιση strog covergece { } lim = =. 2. Σύγκλιση ως προς πιθανότητα covergece i robability lim = 0 lim < =, > 0. { } { } 3. L Σύγκλιση L L covergece { } lim = 0, 0 < <. 4. Σύγκλιση ως προς νόμο (ή κατά κατανομή) covergece i robability law (i distributio) d lim F x = F x, x. Πρόταση: Θα δίξουμ ότι ( ) ( ) L d. { } { } lim lim = 0 2. Έχουμ ότι { io} Θέτουμ B { },.. = 0, > 0. =, τότ ( ) ( ( )) ( ) ( ) ( ) d ( { }) lim B lim J = lim J = if J : Σπυρίδων Ι. Χατζησπύρος Πιθανότητς ΙΙ Ακολουθίς νδχομένων
2 = lim su B = {, io..} = 0 lim { } = 0, > 0 3. Ισχύι ότι { x} { x } νδχομένων { x+ } και { } { x} { x } { } + < + το τλυταίο νδχόμνο, ίναι η τομή των >, και έτσι έχουμ + > που δίνι { } { } { } { } { } x x+ + > x+ + >, ή ότι ( ) ( ) { } F x F x+ + >. Παίρνοντας το limsu της προηγούμνης σχέσης, και πιδή { } { } lim su > = lim > = 0, έχουμ ( ) ( + ) lim su F x F x Ισχύι ότι { x } { x} νδχομένων { x> } και { x} { x } { x} { } + < το τλυταίο νδχόμνο, ίναι η τομή των, και έτσι έχουμ > που δίνι { } { } { } { } { } x x + > x + >. Παίρνοντας το limif της προηγούμνης σχέσης, και πιδή { } { } limif > = lim > = 0, έχουμ ( ) limif ( ) F x F x. Συνολικά λοιπόν ( ) limif ( ) limsu ( ) ( ) F x F x F x F x+, και αν το x ίναι σημίο συνέχιας της F, θα έχουμ ( ) limif ( ) limsu ( ) ( ) F x F x F x F x. Σπυρίδων Ι. Χατζησπύρος Πιθανότητς ΙΙ Ακολουθίς νδχομένων 2
3 Δηλαδή limif F ( x) = F ( x) = limsu F ( x) ή ότι lim F ( x ) F ( x ) =. Lemma: as imlies 0 as m,, or equivaletly { } { m } lim > = 0, > 0 lim > = 0, > 0. m { m > } = { m + m > } { m + > } m > >, > m Takig robabilities { } the limit as m, gives m > m > + > 2 2 ad the lim { m > } lim m > + lim > m 2 2 m { } lim > = 0, > 0 0. m m m Remark: The latter lemma tells us that if a sequece of r.v.s coverges i robability the it is Cauchy i robability. The coverse is also true (theorems 4, 5, ad, 6 Shiryaev AN 259). Theorem (rotter, Jacod J robability Essetials 43) lim = 0. + W.l.o.g. it suffices to show that 0 lim = 0. + ( ) = { > } + { } Σπυρίδων Ι. Χατζησπύρος Πιθανότητς ΙΙ Ακολουθίς νδχομένων 3
4 Whe it is always true that + which is true for +. Whe 0< < we have as. Therefore { } + { } > { > } { > } + lim lim { > } Because 0 we have { } lim = 0. + lim > = 0 ad lim, > 0 + ( ) The fuctio f ( x) x = + x the f ( ) f { } ( ) f { } ( ) > >. Takig exectatios ad the limits yields ( ) { } ( ) f lim > lim f = 0 0. Exercise f : + + : Let be icreasig, bouded, ad cotiuous with ( ) { ( )} lim f = 0. W.l.o.g. it suffices to show that { f ( )} ( ) 0 lim = 0. 0 f ( ) = f ( ) { } + f ( ) > { } ( ) { } ( ) { } { } ( ) ( ) f + f M + f, f x M <, x 0. > > Takig exectatios ad the limits yields { f ( )} M { } f ( ) f ( ) 0 lim lim > + =. f 0 = 0, the Σπυρίδων Ι. Χατζησπύρος Πιθανότητς ΙΙ Ακολουθίς νδχομένων 4
5 The takig the limit as 0 gives { f ( )} f ( ) 0 lim 0 + = 0. ( ) The fuctio ( ) f x the f ( ) f { } ( ) f { } ( ) Takig exectatios ad the limits yields ( ) { } ( ) > > f lim > lim f = x = + x Observe that : The fuctios f ( x), f ( x) = x ad f ( x) arcta ( x) icreasig, bouded, ad cotiuous with f ( 0) = 0. = are Exercise: Show that if ad Y Y as, the + Y + Y as. { ( Y) ( Y) } Y Y ( Y) ( Y) { } + + > = > + > > > > 2 2 { Y Y } Y Y, 0 Takig robabilities ad the the limit as gives { ( + Y) ( + Y) > } > + Y Y > 2 2 lim { ( + Y) ( + Y) > } lim > + lim Y Y > 2 2 { ( ) ( ) } lim + Y + Y > = 0, > 0 + Y + Y.. Σπυρίδων Ι. Χατζησπύρος Πιθανότητς ΙΙ Ακολουθίς νδχομένων 5
6 Assume that a sequece of r.v.s { } i robability, a L limit ( ) ( 2 ) = = =, ( ) ad a have simultaeously a limit, a limit 2 L limit ( 2). The We show that { } { } { } that { > 0} = lim { > }. To see that, let 0 0+ A = { > } = { > } { > } = A = = = 0 > 0 = 0. We observe { } { } { } lim > = > = > 0 = 0. = But { } 2 2 as we have { > 0} = 0. as, ad > > + >, the by takig the limit Σπυρίδων Ι. Χατζησπύρος Πιθανότητς ΙΙ Ακολουθίς νδχομένων 6
7 Σπυρίδων Ι. Χατζησπύρος Πιθανότητς ΙΙ Ακολουθίς νδχομένων 7
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