# Estimating Tail Probabilities of Random Sums of Phase-Type Scale Mixture Random Variables +
## 1 Introduction  
Tail probabilities of random sums have attracted the interest of researchers for decades, as they are important quantities in many fields; for instance:  
* In insurance risk theory, these quantities correspond to the ruin probabilities associated with certain initial capital, and the random sum is a geometric sum of ladder heights (integrated tails of the claim sizes) in a claim surplus process; see, for example, [1]. * In queueing theory, stable single server Markovian queues with service times have a geometric length in equilibrium, and these quantities correspond to the probabilities that an arriving customer must wait longer than a certain time; see, for example, [2]. Estimation from sample data and density approximation with phase-type distributions was considered in [3], and parameter estimation for the class of discrete scaled phase-type distributions was subsequently treated in [4]. Maximum likelihood estimation via an expectation-maximisation (EM) algorithm was exploited in both papers. Estimating tail probabilities of random sums has been extensively investigated both for light- and heavy-tailed summands. Classical methods, including large deviations, saddlepoint approximations, and exponential change of measure, are most commonly used. However, the application of these methods requires the existence of the moment generating function (MGF) of the summand, so are limited to light-tailed cases. In the heavy-tailed setting, subexponential theory provides asymptotic approximations but these offer poor accuracy for moderately large values ([5]). From a rare-event simulation perspective, there are a few efficient estimators proposed in the literature ([6, 7, 8]). Simulation methods for theheavy-tailed case often require the cumulative distribution function (CDF) or probability density function (PDF) of the summand, so these are frequently not easily implementable. In this paper, we consider the problem of efficiently estimating the quantity  
\[\ell(u)=\mathbb{P}(S_{N}>u)\, \tag{1}\]  
where \(S_{N}=Z_{1}+\cdots+Z_{N}\), for large \(u\). Specifically, \(N\) is a discrete light-tailed random variable supported over the positive integers and \(\{Z_{i}\}_{i\in\mathbb{N}}\) is independent of \(N\) and forms a sequence of independent, non-negative, and identically distributed random variables having stochastic representation \(Z_{i}=W_{i}\,X_{i}\), where the random variables \(W_{i}\) and \(X_{i}\) are non-negative and independent of each other. In our setting, the sequence \(\{Z_{i}\}_{i\in\mathbb{N}}\) belongs to the class of \(W\)-mixture of _phase- type distributions_ (PH distributions), which can be represented as a product \(W_{i}\,X_{i}\), where each \(W_{i}\) is a non-negative but otherwise arbitrary random variable and \(X_{i}\) is a PH-distributed random variable. The concept of \(W\)-mixture of PH distributions can be found in [9]. [PERSON] et al. [10] proposed a subclass of \(W\)-mixture of PH distributions to approximate any heavy-tailed distribution. Such a class is very attractive in stochastic modelling because it inherits many important properties of PH distributions--including being dense in the class of non-negative distributions and closed under finite convolutions--while it also circumvents the problem that individual PH distributions are light-tailed, implying that the tail behaviour of a heavy-tailed distribution cannot be captured correctly by PH distributions alone. Reference [11] showed that a distribution in the \(W\)-mixture of PH distributions class is heavy-tailed if and only if the random variable \(W_{i}\) has unbounded support. The CDF and PDF of a distribution in the \(W\)-mixture of PH distributions class are both available in closed form but these are given in terms of infinite dimensional matrices. The problem of approximating tail probabilities of scale mixture of PH-distributed random variables is not easily tractable from a computational perspective. [PERSON] et al.