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5.1. Let ξ1 have negative mean and heavy-tailed distribution. Prove the maximum M has a heavy-tailed distribution too. 5.2.Letξ1 havenegativemeanandalight-taileddistribution.Provethemaximum M has a light-tailed distribution too. Hint: Estimate from above the exponential moments of M via those of Sn. 5.3. Suppose that ξ1, ..., ξn are independent random variables with common exponentialdistribution.Findtheasymptoticbehaviourofthetailprobabilityofthe maximum of sums max k≤n (ξα 1 +...+ξα k ) for (i)α > 1; (ii)α < 0. 5.4. Suppose that ξ1, ξ2, ..., are independent random variables which are uniformlydistributedintheinterval[0,1].For0<α <1andε >0,findtheasymptotic behaviour of the distribution density of the maximum of the sums 11 1  max +...+ −n +ε . 1−αα αn ≥1 ξ ξn 1 5.5. Let X(t) be a compound Poisson process with a subexponential distribution for a typical jump. For every T, find the tail asymptotics for the distribution of the supremum sup X(t) 0≤t≤T in terms of the jump distribution. 5.6. Suppose X(t) is a compound Poisson process with a negative drift −a and withajumpdistributionF.Findthetailasymptoticsforthedistributionofthesupremum supX(t) t≥0 provided F is regularly varying at infinity with index α > 0. Find what power moments of this supremum are finite and which are infinite. 5.7. Moving overages. Suppose η , η , η , ...are independent identically dis0 1 2 tributed random variables with common distribution F and zero mean. Let ξ := n η +η −awherea>0.DefineS :=0,S :=ξ +...+ξ andM:=max{S ,n≥ n−1 0 1n n n n 0}. Prove that M is finite with probability 1 and that (i) if the integrated tail distribution F is long

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All HI6008 Students Enrol in the Semester 2/2017 need to follow below Assignment structure: 1. Introduction2. Project Objective 3. Project Scope4. Literature Review (Students’ needs to summarise Assignment 1 literature review (2-3 pages) and justification from Assignment 1 literature prob ...

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Assignment Part1: Estimation and Forecast You have been retained by the fixed income desk of Baek-Brooklyn Investment Group to provide a forecast about future short term interest rates, namely, the 3 month t-bill rate. You decide to use two sources of data: historical interest rate data and curren ...

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