3.13 Problems 3.1. Prove the Cauchy distribution is subexponential and strong subexponential. 3.2. Using direct estimates for convolution of the Pareto density, show the Pareto distribution is subexponential and strong subexponential. 3.3. Using direct estimates for convolutions, prove that any regularly varying at infinity distribution is subexponential and strong subexponential. 3.4. Let F and G be two distributions that are regularly varying at infinity, and let 0 < p < 1. (i) Prove that the distribution pF +(1− p)G is regularly varying at infinity. (ii) Prove that the distribution F ∗G is regularly varying at infinity. 3.5. Prove by direct calculations of convolution density that the exponential distribution is not subexponential. 3.6. Suppose that x1, . . . , xn are independent random variables with common distribution F. Prove the maximum, max(x1;:::;xn), has subexponential distribution if and only if F is subexponential. 3.7. Suppose that F and G are weakly tail-equivalent distributions on R+. Prove that the convolutions F ∗F and G∗G are weakly tail-equivalent too. Prove the same for n-fold convolutions, for any n ≥ 3. 3.8. Suppose that x1, . . . , xn are independent random variables with a common a1 exponential distribution. Find the asymptotics for probability Pfx1 + ::: + xn an > xg as x ! ¥ if (i) all ai > 1; (ii) all ai < 0; (iii) all ai 62 [0;1]. 3.9. Let non-negative random variable x have the Weibull distribution with the b tail F(x) = e−x , b > 0. For which values of a > 0, does Pfx +x a > xg ∼ Pfx > xg as x ! ¥? Hint: Make use of the equality x + x a = x a(1+ x 1−a) and Problem 2.23. 3.10. Let random variable x have the standard log-normal distribution. For which values of parameter a > 0, does Pfx + x a > xg ∼ Pfx > xg as x ! ¥? 3.11. How do Pitman's criteria work (i) for the Pareto distribution; (ii) for the exponential distribution? 3.12. Specify Kesten's bound for the standard Cauchy distribution. 3.13. Let X(t) be a compound Poisson process with a subexponential distribution of jump. For every t, find the asymptotic tail behaviour of the distribution of X(t) in terms of the jump distribution. 3.14. Find an example of subexponential distribution F and of counting random variable t such that the equivalence F∗t(x) ∼ EtF(x) as x ! ¥ doesn't hold. Hint: Make a link to Galton-Watson process.3.15. Markov modulated random walk. Suppose Xn is a time-homogeneous Markov chain with state space f1;2g and transition probabilities pi j. Suppose xn, n ≥ 0, are independent identically distributed random variables with common subexponential distribution F and hn, n ≥ 0, are also independent with common subexponential distribution G and such that x 's and h's are mutually independent. Assume zn = xn if Xn = 1 and zn = hn if Xn = 2. Assume that G(x) ∼ cF(x) as x ! ¥, with 0 ≤ c < ¥. (i) Find the tail asymptotics for the distribution of z0 + z1 in terms of F. (ii) Denote t1 := minfn > 0 : Xn = X0g. Find the tail asymptotics for the distribution of z0 + :::+ zt1 in terms of F. (iii) For k ≥ 2, let tk = minfn > tk−1 : Xn = X0g. Find the tail asymptotics for the distribution of z0 + :::+ ztk in terms of F. 3.16. Suppose F is long tailed distribution on R+ such that the corresponding integrated tail distribution FI is subexponential. Prove that (F ∗F)I is subexponential too. Hint: Make use of the equality in Problem 2.22. 3.17. Suppose x1, . . . , xn are independent nonnegative random variables with common subexponential distribution F. (i) Prove that Pqx1 2 + :::+ xn 2 > x ∼ nF(x) as x ! ¥: (ii) More generally, prove that, for any convex function g : R+ ! R+, Pfg−1(g(x1)+ :::+ g(xn)) > xg ∼ nF(x) as x ! ¥: 3.18. Let random variable x have an intermediate regularly varying distribution and random variable h a light-tailed distribution. Prove that, for any joint distribution of x and h, Pfx + h > xg ∼ Pfx > xg as x ! ¥: 3.19. Let h be a positive random variable and x1, x2 be two identically distributed random variables which are conditionally independent given any value of h, that is, a.s. Pfx1 2 B1;x2 2 B2 j hg = Pfx1 2 B1 j hgPfx2 2 B2 j hg for all Borel sets B1 and B2. Find the exact asymptotics for Pfx1 > xg and Pfx1 + x2 > xg in the following cases: (i) Pfxi > x j hg = (1 + x)−h a.s. and h is uniformly distributed in the interval [1;2]. Hint: Make use of the following bounds: Pfx1 + x2 > x) ≥ Pfx1 > xg+Pfx2 > xg −Pfx1 > x;x2 > xg and, for any function h(x) < x=2, Pfx1 + x2 > xg ≤ Pfx1 > x − h(x)g+Pfx2 > x − h(x)g +Pfh(x) < x1 < x − h(x);x2 > x − x1g: h (ii) Pfxi > x j hg = e−x and h is uniformly distributed in the interval [1=2;3=2]. h (iii) Pfxi > x j hg = e−x and h is uniformly distributed in the interval [0;1]. 3.20. Let x1, x2, h1, h2 be four mutually independent random variables where x1 and x2 have a regularly varying distribution with parameter a > 0 while h1 and h2 have a uniform distribution in the interval [−1;1]. Find the asymptotics, as x ! ¥, for the following probabilities: (i) Pfx1eh1 + x2eh2 > xg; (ii) Pfx1eh1 + x2eh1+h2 > xg. 3.21. Excess process. In conditions of Problem 2.26, prove that the invariant distribution is subexponential if and only if the integrated tail distribution FI is subexponential. 3.22. In Problem 2.27, introduce extra conditions that are sufficient for the subexponentiality of the invariant distribution.