mat280-hw1

# mat280-hw1 - MAT 280 UC Davis Winter 2011 Longest...

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MAT 280 — UC Davis, Winter 2011 Longest increasing subsequences and combinatorial probability Homework Set 1 Homework due: Wednesday 2/2/11 1. Let Λ = lim n →∞ n / n as in Theorem 6 in the lecture notes. The goal of this problem is to show that the bounds 1 Λ e that follow from Lemmas 3 and 4 can be improved to (8 ) 1 / 2 Λ 2 . 49. (a) In the proof of Lemma 4 observe that if L ( σ n ) t then X n,k ( t k ) , so the bound in (1.4) in the notes can be improved. Take k α n and t β n and optimize the improved bound over α < β (using some version of Stirling’s formula) to conclude that Λ 2 . 49. (b) Given a standard Poisson Point Process (PPP) Π in [0 , ) × [0 , ), con- struct an increasing subsequence ( X 1 ,Y 1 ) , ( X 2 ,Y 2 ) , ( X 3 ,Y 3 ) ,... of points from the process by letting ( X 1 ,Y 1 ) be the Poisson point that minimizes the coordi- nate sum x + y , and then by inductively letting ( X k ,Y k ) be the Poisson point in ( X k - 1 , ) × ( Y k - 1 , ) that minimizes the coordinate sum x + y . Observe that the properties of the Poisson point process imply that one can write ( X k ,Y k ) = k X j =1 ( W j ,Z j ) , where ( ( W j ,Z j ) ) j =1 is a sequence of independent and identically distributed random vectors in [0 , ) 2 , each having the same distribution as ( X 1 ,Y 1 ). Com- pute the expectations μ X = E ( X 1 ) Y = E ( Y 1 ) . Hint:

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mat280-hw1 - MAT 280 UC Davis Winter 2011 Longest...

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