Problems 2.1. Let distribution F on R+ has a regularly varying tail with index α, i.e., let 2 F(x) = L(x)/xα where function L(x) is slowly varying at infinity. Prove that: (i) Any power moment of distribution F of order γ < α is finite. (ii) Any power moment of order γ > α is infinite. Show by examples that the moment of order γ = α may either exist or not, depending on the tail behaviour of slowly varying function L(x). 2.2. Let distribution F on R+ have a regularly varying tail with index α > 0. Prove the distribution of logξ is light-tailed. 2.3. Let ξ > 0 be a random variable. Prove that the distribution of logξ is lighttailed if and only if ξ has a finite power moment of order α, for some α > 0. 2.4. Let distribution F on R+ have an infinite moment of order γ > 0. Prove that F is heavy-tailed. 2.5. Let random variable ξ ≥ 0 be such that Eeξ α = ∞ for some α < 1. Prove that the distribution of ξ is heavy-tailed. 2.6. Let random variable ξ has (i) exponential; (ii) normal distribution. Prove that the distribution of eαξ is both heavy- and long-tailed, for every α > 0. 2.7. Student's t-distribution. Assume we do not know the exact formula for its density. By estimating the moments, prove that the distribution of the ratio ξ (ξ1 2 + ...+ ξn 2)/n is heavy-tailed where the independent random variables ξ , ξ1, . . . , ξn are sampled from the standard normal distribution. Moreover, prove that this distribution is regularly varying at infinity. Hint: Show that the denominator has a positive density function in the neighbourhood of zero. 2.8. Let η1,...,ηn be n positive random variables (we do not assume their independence, in general). Prove that the distribution of η1 + ... + ηn is heavy-tailed if and only if the distribution of at least one of the summands is heavy-tailed. 2.9. Let ξ > 0 and η > 0 be two random variables with heavy-tailed distributions. Can the minimum min(ξ ,η) have a light-tailed distribution?2.10. Suppose that ξ1, . . . , ξn are independent random variables with a common distribution F and that ξ(1) ≤ ξ(2) ≤ ... ≤ ξ(n) are the order statistics. (i) For k ≤ n, prove that the distribution of ξ(k) is heavy-tailed if and only if F is heavy-tailed. (ii) For k ≤ n − 1, prove that the distribution of ξ(k+1) − ξ(k) is heavy-tailed if and only if F is heavy-tailed. (iii) Based on (ii) and on Problem 8, prove that ξ(k) − ξ(l) has a heavy-tailed distribution if and only if F is heavy-tailed. 2.11. Let ξ and η be two positive independent random variables. Prove that the distribution of ξ − η is heavy-tailed if and only if the distribution of ξ is heavytailed. 2.12. Let ξn, n= 1, 2,. . . , be independent identically distributed random variables on R+. Let ν ≥ 1 be an independent counting random variable. Let both ξ1 and ν have light-tailed distributions. Prove that the distribution of random sum ξ1 + ξ2 + ...+ ξν is light-tailed too. 2.13. Let ξn, n = 1, 2, . . . , be independent identically distributed random variables on R+ such that P{ξ1 > 0} > 0. Let ν ≥ 1 be an independent counting random variable. Let ν have heavy-tailed distribution. Prove that the distribution of random sum ξ1+ ξ2+ ...+ ξν is heavy-tailed. 2.14. Find a light-tailed distribution F such that the distribution of the product ξ1ξ2 is heavy-tailed where ξ1 and ξ2 are two independent random variables with distribution F. 2.15. Let non-negative random variable ξ has distribution F. Consider a family of distributions Fx(B) := P{ξ ∈ x+ B|ξ > x}, B ∈ B(R+). (i) Prove F is long-tailed if and only if Fx ⇒ ∞, as x → ∞. (ii) Prove F is h-insensitive if and only if ξx/h(x) ⇒ ∞ as x → ∞ where ξx is a random variable with distribution Fx. 2.16. We say that H(x) is a boundary function for a long-tailed distribution F if the following condition holds: F is h-insensitive if and only if h(x) = o(H(x)) as x → ∞. Find any boundary function for: (i) A regularly varying distribution with index α > 0. (ii) A standard log-normal distribution. (iii) A Weibull distribution with tail F(x) = e−xβ where 0 < β < 1. (iv) A distribution with tail F(x) = e−x/log(1+x), x ≥ 0. 2.17. Prove that a distribution whose tail is slowly varying at infinity does not have a boundary function. 2.18. Let a random variable ξ has the standard normal distribution. (i) Find all values of α > 0 such that the power |ξ|α has a heavy-tailed distribution. (ii) Prove the power |ξ|α has a heavy- and long-tailed distribution for every α < 0.2.19. Let independent random variables ξ1, . . . , ξn have the standard normal distribution. Find all n such that the product ξ1 · ... · ξn is heavy-tailed. For those n, is the product also long-tailed? 2.20. Let independent non-negative random variables ξ1, . . . , ξn have Weibull distribution with the tail F(x) = e−xβ , β > 0. Find the values of β , for which the product ξ1 · ...· ξn has a heavy-tailed distribution. 2.21. Perpetuity. Suppose ξ1, ξ2, . . . are independent identically distributed random variables with common uniform distribution in the interval [−2,1]. Let S0 = 0, Sn = ξ1+ ...+ ξn and Z = ∞∑n=0 eSn. (i) Prove Z is finite with probability 1 and that Z has a heavy-tailed distribution. Hint: Show that EZγ = ∞ for some γ > 0. (ii) How can the result of (i) be generalised to other distributions of ξ 's? 2.22. Let F and G be two distributions on R+ with finite means aF and aG. Prove that, for all sufficiently large x, (F ∗ G)I(x) = FI(x)+ F ∗ GI(x)+(aG − 1)F(x) = GI(x)+ FI ∗ G(x)+(aF − 1)G(x). 2.23. Prove the distribution of a random variable ξ ≥ 0 is √x-insensitive if and only if the distribution of ξ is long-tailed. More generally, prove that the distribution of ξ ≥ 0 is xα-insensitive with some 0 < α < 1 if and only if the distribution of ξ1−α is long-tailed. 2.24. Suppose Xn is a time-homogeneous Markov chain with state space Z+ and transition probabilities pi j. Let Xn be a skip-free Markov chain, i.e., only the transition probabilities pi,i−1, pi,i and pi,i+1 are non-zero. Let Xn be positive recurrent with invariant probabilities πi. Show that πi = i∏j=1 p j−1, j p j, j−1 . Further, assume that limsupi→∞ pii < 1. (i) Prove if limsupi→∞(pi,i+1 − pi,i−1) < 0, then the invariant distribution is lighttailed. (ii) Prove if limsupi→∞(pi,i+1 − pi,i−1) = 0, then the invariant distribution is heavytailed.