**4. Asymptotic Approximation**

In many cases, it is of importance to investigate the asymptotic behaviour of the ruin probability when the initial risk reserve tends to infinity. This question is particularly interesting in the case of heavy-tailed claim sizes. Towards this direction, when the claim sizes belong to the class of subexponential distributions S (Teugels 1975), e.g., Pareto, Weibull, Lognormal, etc., the following asymptotic approximation is classical (see, e.g., Embrechts and Veraverbeke 1982):

**Theorem 3.** *Suppose in the general Sparre Andersen model that the claim sizes and interclaim times have both finite means* E*X and* E*W, respectively, such that c*E*W* > E*X. If* 1E*X u*0*FX*(*x*)*dx* ∈ S*, then*

$$\psi(u) \sim \psi\_{\mathcal{S}}(u) := \frac{1}{c \mathbb{E} \mathcal{W} - \mathbb{E}X} \int\_{u}^{+\infty} \mathbb{F}\_{X}(x) dx, \qquad \text{as } u \to +\infty.$$

Note that the heavy-tail approximation *ψ*S (*u*) holds for any interclaim time distribution. However, further modifications have been attained in Willmot (1999), when the Laplace transform of the interclaim times is a rational function of the form (5) with *β*(*s*) = *β* and *FX* belongs to the subclass of regularly varying distributions, i.e., *FX*(*u*) ∼ *<sup>L</sup>*(*u*)*u*<sup>−</sup>*α*−1*e*<sup>−</sup>*γ<sup>u</sup>*, *u* → +<sup>∞</sup>, where *<sup>L</sup>*(*u*) a slowly varying function and *α* > 0, *γ* ≥ 0. For example, the Pareto(*<sup>a</sup>*, *b*) distribution (see Section 5.2.1) belongs to the class of regularly varying distributions with *<sup>L</sup>*(*u*) = *b* + 1/*u*−*a*, *α* = *a* − 1 and *γ* = 0, and its modified asymptotic approximation is then given by

$$\psi(u) \sim \psi\_{\mathcal{M}}(u) := \frac{L(u)u^{-a}}{a(c \to W - \mathbb{E}X)} = \frac{(1+bu)^{-a+1}}{(a-1)\left(b+\frac{1}{u}\right)(c \to W - \mathbb{E}X)},$$

which is smaller than *ψ*S (*u*) by a factor *bu bu* + 1 that converges to 1 as *u* → +∞; see Willmot (1999) for details.

Clearly, the heavy-tail approximation admits a simple formula whenever the expectations of the interclaim times and claim sizes are finite; however, it has a drawback that occurs when *c*E*W* ≈ E*X* the approximation is useful only for extremely large values of *u*.

In the next section, we compare the accuracy of the spectral approximation to the accuracy of the heavy tail one, i.e., *ψ*S (*u*). An interesting observation is that the spectral approximation converges faster to zero than any heavy-tailed distribution due to the exponential decay rate of the former. Thus, the heavy-tail approximation is expected to outperform the spectral approximation in the far tail, but for medium values, this new approximation can be very competitive.
