2. HEAVY-TAILED DISTRIBUTIONS

The most important class of heavy-tailed distributions is the subexponential class. By definition, a distribution F = 1 − F on [0,∞) or its corresponding random variable is said to be subexponential, denoted by

, if the relation

holds for some (or, equivalently, for all) n = 2,3,…, where F*ⁿ denotes the n-fold convolution of F. More generally, a distribution F on (−∞,∞) is still said to be subexponential if the distribution F⁺(x) = F(x)1_(x≥0) is subexponential. By Lemma 2.1 and the last inclusion of (2.3) below, it is easy to verify that (2.1) remains valid for the latter general case. The class

contains the Pareto-like, the lognormal-like, and the Weibull-like distributions.

Closely related are the class

of long-tailed distributions and the class

of distributions with dominatedly varying tails. A distribution F on (−∞,∞) belongs to the class

if the relation

holds for some (or, equivalently, for all) y > 0; F belongs to the class

if the relation

holds for some (or, equivalently, for all) 0 < y < 1.

It is well known that

For more details of heavy-tailed distributions, we refer the reader to Embrechts, Klüppelberg, and Mikosch [8] and references therein.

In what follows, all limiting relationships are for x → ∞ unless stated otherwise. For two positive functions a(·) and b(·), we write a(x) = O(b(x)) if lim sup a(x)/b(x) < ∞, a(x) = o(b(x)) if lim a(x)/b(x) = 0, a(x) ∼ b(x) if lim a(x)/b(x) = 1, and a(x) [lsim ] b(x) if lim sup a(x)/b(x) ≤ 1.

The following lemma is well known; see Embrechts and Goldie [7], Cline [5, Cor. 1], and Tang and Tsitsiashvili [17, Lemma3.2].

Lemma 2.1: Let F be the convolution of two distributions F₁ and F₂. If

, then

and

The following lemma is from Cline and Samorodnitsky [6, Thm.2.1].

Lemma 2.2: Let X and Y be two independent random variables with distributions F and G, respectively, satisfying

. The distribution H of the product XY is subexponential if there is a positive function a(x) = o(x) such that F(x − a(x)) ∼ F(x) and G(a(x)) = o(H(x)).

3. FOR THE STANDARD CASE

Let us go back to the model introduced in Section 1. Hereafter, denote the generic random variable of {X_n : n = 1,2,…} (under assumption P1) by X, the generic random variable of {Y_n : n = 1,2,…} (under assumption P2) by Y, and the distribution of XY (under assumptions P1, P2, and P3) by H = F [otimes ] G.

The main result of Tang and Tsitsiashvili [17] is that, under assumptions P1, P2, and P3, the relation

holds for each n = 1,2,… if

for some large p > 0 (more precisely, for some p larger than the upper Matuszewska index of the distribution F). The estimate given by (3.1) enables us to recursively calculate the ruin probability ψ(x,n). However, an obvious drawback is that the condition

excludes many popular distributions such as the lognormal-like and the Weibull-like distributions; recall (2.2).

The following is our first main result, which extends the scope of the loss distribution to the whole subexponential class

:

Theorem 3.1: Assume P1, P2, and P3. If

and there is some 0 < τ < 1 such that F(x − x^τ) ∼ F(x) and G(x^τ) = o(H(x)), then (3.1) holds for each n = 1,2,….

Two concrete cases of Theorem 3.1 are listed below without proof.

Corollary 3.1: Assume P1, P2, and P3. Relation (3.1) holds for each n = 1,2,… if one of the following groups of conditions is valid:

Clearly, both in Theorem 3.1 and Corollary 3.1, the condition G(x^τ) = o(H(x)) is implied by G(x^τ) = o(F(x)). More concretely, in Corollary 3.1(A) the G can be every Weibull distribution or every lognormal distribution with a density function

as long as

, and in Corollary 3.1(B) the G can be every Weibull-like distribution with a tail

as long as

.

In the proof of Theorem 3.1 we will need the following lemma.

Lemma 3.1: Under the conditions of Theorem 3.1, it holds for each k = 1,2,… that

Proof: We only prove the result for k = 1 since the general case extends by induction. Trivially, the condition F(x − x^τ) ∼ F(x) implies that F(x − Cx^τ) ∼ F(x) holds for every constant C > 0. Choose some 0 < ε < 1 such that G(ε) > 0. Then it holds for all large x > 0 and t ∈ (ε,x^τ] that

For all large x > 0, we derive

It follows that

Since G(0) = 0 and ε > 0 can be arbitrarily small, we obtain

which actually amounts to H(x − x^τ) ∼ H(x). █

Proof of Theorem 3.1: Recall the two-sided inequality (1.3). If we can prove the relation

without using F(0−) > 0, then the same proof should also be valid for the relation

and we immediately obtain (3.1). Write

Under assumptions P1, P2, and P3, it is clear that

where =^d denotes “equal in distribution.” Based on this analysis, it suffices to prove the relation

In view of Theorem 4.1 below, we only need to consider the case that Y is unbounded. We prove the asymptotic relation (3.2) by the inductive method.

Trivially, (3.2) holds for n = 1. Applying Lemma 2.2, we also know that V₁ is subexponential. Now we assume by induction that (3.2) holds for n = m ≥ 1 and that V_m is subexponential. Clearly, F(x) = O(Pr(V_m > x)) since G(1) > 0. From Lemmas 2.1 and 3.1 and the inductive hypothesis, it follows that the sum X_m+1 + V_m is subexponential and that

Hence, by Lemma 2.2, the random variable V_m+1 is subexponential. By Lemma 2.1 and the inductive hypothesis, we derive that

This proves that (3.2) holds for n = m + 1. By the mathematical inductive method, we conclude that (3.2) holds for each n = 1,2,…. █

4. FOR THE CASE OF ASSOCIATED DISCOUNT FACTORS

Recently, the study on ruin probabilities of nonstandard models has become an important part of risk theory. We refer the reader to Cai [1,2] and Cai and Dickson [4], among many others.

Now we propose a general (positively) dependence structure for the discount factors. We say that a sequence of random variables {Y_n : n = 1,2,…} is (positively) associated if the inequality

holds for all n = 1,2,… and all coordinatewise (not necessarily strictly) increasing functions f₁ and f₂ for which the moments involved exist. Since it was introduced by Esary, Proschan, and Walkup [9], this dependence structure has been extensively studied and applied by many researchers in statistics, applied probability, insurance, and finance. Trivially, if in the above definition f₁ is coordinatewise increasing but f₂ is coordinatewise decreasing, then (4.1) is changed to

The following is our second main result, which partially extends Theorem 3.1 to the proposed nonstandard case:

Theorem 4.1: Assume P1, P3, and

If

, then (3.1) holds for each n = 1,2,….

Theorem 4.1 indicates that the association of the bounded discount factors does not influence the asymptotic relation (3.1). Moreover, if we restrict the discussion to the case of Pareto-like loss distributions, then under assumptions P1 and P3, using a result of Resnick and Willekens [16], it is not difficult to prove that (3.1) even holds for arbitrarily dependent discount factors {Y_n : n = 1,2,…} as long as they satisfy suitable summability conditions.

We also remark that the boundedness condition of Theorem 4.1 is not so restrictive for application. For example, it allows for a realistic case below (see also Example 4.1 of Tang and Tsitsiashvili [19]).

Suppose that an insurer invests his wealth not only in a risk-free asset (a bank) but also in a risky asset (a stock market). At time n − 1, the insurer has wealth S_n−1, and he keeps a nonrandom fraction, say 0 < a_n ≤ 1, of his wealth in the bank and invests the remaining part in the stock market. Then, at time n, the first part becomes a_n(1 + r_n)S_n−1 with some deterministic interest rate r_n ≥ 0 and the second part becomes (1 − a_n)(1 + R_n)S_n−1 with some stochastic return rate R_n ∈ [−1,∞). Consequently, the discount factors equal

which are obviously bounded from above by positive constants.

For related discussions in continuous-time settings, see Hipp and Plum [12], Gaier and Grandits [10], Gaier, Grandits, and Schachermayer [11], Cai [3], Liu and Yang [13], among others.

Additionally, from (4.3) we see that if 0 < a_n < 1 for n = 1,2,…, the association of {Y_n : n = 1,2,…} is equivalent to that of {R_n : n = 1,2,…}.

In the proof of Theorem 4.1, we will need the following lemma, which is a restatement of Proposition 5.1 of Tang and Tsitsiashvili [18].

Lemma 4.1: Let {X₁,…,X_n} be n i.i.d. real-valued random variables with common distribution

. Then, for arbitrarily fixed 0 < a ≤ b < ∞, the relation

holds uniformly for (c₁,…,c_n) ∈ [a,b] × ··· × [a,b]; that is,

Proof of Theorem 4.1: Choose some constant d = d_n > 1 as a common upper bound of the random variables {Y₁,…,Y_n}. First, we derive an asymptotic upper bound for ψ(x,n). For an arbitrarily fixed 0 < ε < 1 such that Pr(ε < Y_j ≤ d) > 0 for each j = 1,…,n, we split the probability on the right-hand side of (1.3) into two parts as

Since the random variables {Y₁,…,Y_n} are associated and are independent of the nonnegative random variables {X₁⁺,…,X_n⁺}, by (4.2) it holds that

Substituting (4.6) into (4.5) and rearranging the resulting inequality, we have

For J₂(x,ε), on the event

, we have

Hence, by Lemma 4.1 and the independence between {X₁,…,X_n} and {Y₁,…,Y_n},

Substituting (4.8) into (4.7) yields

Since the random variables {Y₁,…,Y_n} are positive and ε > 0 can be arbitrarily small, we prove that

Now we aim at an asymptotic lower bound. From (1.3), Lemma 4.1, and the association of the random variables {Y₁,…,Y_n}, we derive

As earlier, by letting ε [nearr ] 0, we conclude that

Combining (4.9) and (4.10) leads to the announced result (3.1). █