b.solns.billingsley

b.solns.billingsley - Probability II MAP6473 4810 Solns-B...

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Unformatted text preview: Probability II MAP6473 4810 Solns-B Prof. JLF King 4May2010 Tools. For each speed s R , use E s as a name for the map z 7 e i sz on R . Note that | e i - 1 | 6 | | , for all real . This since chord 6 arc. 1 : By the way, unmarked integrals Z shall mean Z R . 2 : Equi-cts Lemma. On compact metric space J we have real-valued functions h n g pointwise . The convergence will be uniform , if { h n } n is a uniformly equi-continuous family. Proof. The limit g automatically has the same ,-relation as the family; so we may replace each h n by h n- g . I.e, WLOGenerality The ( h n ) n converge pointwise to zero . Given , take the corresponding from the family. Now pick a -dense set, F , of points in J ; we can take F finite , since J is compact. Discarding the first few h n , we now have n, t F : | h n ( t ) | 6 ; this, by the convergence to zero. For an arbitrary point s J there is a point t F which is -close to s . Their h n-values are thus -close. The upshot is that for all n : | h n ( s ) | 6 2 . Consequently, back in our original notation we have that k h n- g k sup 6 2 , for all large n . 3 : Bnd Lemma. On prob.space ( , ) , meas.map f : R has f () 6 7 . If R f d > 7 then f () a.e = 7 . Proof. As { f < 7 } = S j { f 6 7- 1 j } , I need but fix a number L< 7 and show that v := ( f () 6 L ) is zero. But R f 6 [1- v ] 7 + v L = 7- v [7- L ]. So v = 0. B1 : For a prob.meas. , let be flipped; so ( B ) := (- B ) . Write the char.fnc ITOf (In Terms Of) . Let h ;7 i and h ;7 , 3 i be a translation and a translation-scaling of : h ;7 i ( B ) := ( B- 7); h ;7 , 3 i ( B ) := (3 B- 7) ....
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This note was uploaded on 01/26/2012 for the course MAP 6473 taught by Professor King during the Fall '11 term at University of Florida.

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b.solns.billingsley - Probability II MAP6473 4810 Solns-B...

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