2.1. Let distribution F on R+ has a regularly varying tail with index α ,i.e.,let 2 F(x)=L(x)/x α where function L(x) is slowly varying at infinity. Prove that: (i) Any power momentof distribution F of...

2.1. Let distribution F on R+ has a regularly varying tail with index
α ,i.e.,let
2
F(x)=L(x)/x α
where function L(x) is slowly varying at infinity. Prove that: (i) Any power momentof distribution F of order γ < α="" is="" finite.="" (ii)="" any="" power="" moment="" of="" order="" γ=""> α is infinite. Show by examplesthat the momentof order γ = α may either exist or not, depending on the tail behaviourof slowly varyingfunctionL(x). 2.2. Let distribution F on R+ have a regularly varying tail with index α > 0. Prove the distribution of log ξ is light-tailed. 2.3. Let ξ > 0 be a random variable. Prove that the distribution of log ξ is lighttailed if and only if ξ has a finite power moment of order α ,forsome 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 ≥0besuchthatEe ξ 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="" e="" αξ="" 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


( ξ
2 1 +...+ ξ
2 n)/n
is heavy-tailed where the independent random variables ξ
, ξ
1,..., ξ
n 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 η 1,..., η n be n positive random variables (we do not assume their independence, in general). Prove that the distribution of η 1 +...+ η n is heavy-tailed if and only if the distribution of at least one of the summandsis heavy-tailed. 2.9.Let ξ >0and η >0betworandomvariableswithheavy-taileddistributions. Can the minimummin( ξ , η ) have a light-tailed distribution