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Sheet6

Sheet6 - 1 Â by Ïˆ Ïˆ âˆ’ 1 t = t for all t Suppose X i i...

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Statistics 330/600 Due: Thursday 24 February 2005: Sheet 6 *(6.1) Find a probability measure on a set Ä with only four points, and sets A i Ä such that P ( A 1 A 2 A 3 ) = ( P A 1 )( P A 2 )( P A 2 ) but for which the sigma-±elds G i : = σ { A i } ,fo r i = 1 , 2 , 3, are not independent. [Can you also do it with an Ä consisting of only three points?] *(6.2) (2 k th moment, with orphans dying) UGMTP Problem 4.24. *(6.3) Let ψ : R + R + be a continuous, strictly increasing function with ψ( 0 ) = 0 and ψ( t ) ↑∞ as t ↑∞ . De±ne ψ
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Unformatted text preview: 1 ( Â· ) by Ïˆ ( Ïˆ âˆ’ 1 ( t ) ) = t for all t . Suppose { X i : i âˆˆ N } is a sequence of independent, identically distributed random variables for which Ïˆ( | X 1 | ) is integrable. Show that P { Ï‰ : | X i (Ï‰) | â‰¥ Ïˆ âˆ’ 1 ( i ) for inÂ±nitely many i } = . (6.4) (Etemadi SLLN) UGMTP Problem 4.26. I would encourage you to work on this problem in groups of more than one student....
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