The collective model of risk theory consists of a random variable \( N \) with \( P_N[{\mathbb N_0}]=1 \) and a sequence of random variables \( \{Y_i\}_{i\in\mathbb N} \) which is assumed to be iid and independent of \( N \). In actuarial mathematics, \( N \) is the number of claims in a portfolio of risks, \( Y_i \) is the severity of claim \( i \), and \( S := \sum_{i=1}^N Y_i \) is the aggregate claim. Since \( P_S[B] = \sum_{n=0}^\infty P_N[\{n\}]\,P_Y^{\ast n}[B] \), the compound distribution \( P_S \) is a mixture of the convolution powers of \( P_Y \) with mixing distribution \( P_N \). This book gives a comprehensive introduction to the recursive computation of convolutions and compound distributions.
Since the claim severity distribution may be degenerate with \( P_Y[\{1\}]=1 \), every recursion for \( P_S \) contains a recursion for \( P_N \); also, if \( P_Y[{\mathbb N_0}]=1 \), then the probability generating functions are related by the identity \( m_S(t)=m_N(m_Y(t)) \) which may used to extend a recursion for \( N \) to a recursion for \( S \) (although this is not the method preferred by the authors). Therefore, it is not surprising that the major part of the book is devoted to recursions for \( P_N \). In particular, the authors study the quite general case where the identity \[ P[\{N=n\}] = \sum_{j=1}^k \biggl( a_j + \frac{b_j}{n} \biggr)P[\{N=n-j\}] \] holds for all \( n \geq l \) with \( k,l\in\mathbb N \) (which for \( k=l=1 \) defines the famous Panjer class including the Poisson distribution) and they also study mixed Poisson distributions and their connection with infinitely divisible distributions.
Further topics include recursions for convolutions and moments, recursions for the aggregate claim in the individual model (in which the claim number is a constant and the claim severities are independent but not necessarily identically distributed), approximations of the aggregate claims distribution in the individual model by that of an appropriate collective model, and certain extensions to the case where either the claim severities or the claim number are multivariate. A particularly interesting aspect of this book is the systematic use of the DePril transform.
Overall, the book provides considerable insight into recursions for convolutions and compound distributions. Whether the results are useful in actuarial practice has to be decided by actuaries: Panjer’s recursion for compound Panjer distributions was revolutionary about 30 years ago and since then computational power has increased tremendously. Nevertheless, the literature on numerical stability and efficiency of recursive methods seems to be scarce, and this topic is also beyond the scope of this book.
Reviewer’s remarks: (1) Because of the style in which the book is written, the reader must be patient: For example, the authors use a sophisticated notation which makes it necessary to consult the list of symbols more frequently than usual, and instead of repeating a formula they usually refer to it by its number; also, the search for a particular result is rather time consuming since e.g. the number 2.1 is used not only for a section but also for a lemma, a theorem, and a table. In wide parts of the book, the reader is expected to accept manipulations of formulas without knowing the aim of the exercise; an extreme example is Chapter 20 which contains 68 numbered formulas but very few explicit results (whose proofs are short). (2) At several occasions, the authors use identities for conditional expected values which are neither evident nor proved. (3) In the discussion of mixed and compound Poisson distributions, the vast class of Hofmann distributions is mentioned only in a comment. (4) The DePril transform is defined in several steps with increasing generality. Since this is a central tool in the book, it would be preferable to start with the general definition and then demonstrate the power of the transform.