Many probabilistic models estimating the likelihoods of certain events are based on the sequence of sums { ∑ i = 1 n X i , n ∈ N }, where X i are some random variables. This sequence is called the random walk. Random walks are usually visualized as branching trees or certain paths on the plane or space; their occurrence spreads from pure mathematics to many applied sciences. For instance, one may refer to the Case–Shiller home pricing index [9] or even more generally to the random walk hypothesis [30]. From a pure mathematics standpoint, one may mention the random matrix theory, see, for example, [3, 13, 31] and other related works. Perhaps the closest context where the need to know the distribution of sup n ⩾ 1 ∑ i = 1 n ( X i − κ ) arises is insurance mathematics. In the ruin theory one may assume that the insurer’s wealth W u ( n ) in discrete time moments n ∈ N consists of incoming cash premiums and outgoing payoffs (claims), and W u ( n ) admits the representation: (1) W u ( n ) = u + κ n − ∑ i = 1 n X i = u − ∑ i = 1 n ( X i − κ ) , where u ∈ N 0 : = N ∪ { 0 } is interpreted as initial surplus W u ( 0 ) : = u, κ ∈ N, denotes the arrival rate of premiums paid by customers and the subtracted sum of random variables represents claims. Here we assume that random variables X i are independent, nonnegative, and integer-valued but not necessarily identically distributed. More precisely, we assume that X i = d X i + N for all i ∈ N and some fixed N ∈ N. The model (1) can be visualized as a “race” between deterministic line u + κ n and the sum of random variables ∑ i = 1 n X i when n varies, see Figure 1.1¹

Implemented using the RandomChoice function in Wolfram Mathematica [22].

We say that the fixed natural number N is the number of periods or seasons and call the model (1) N-seasonal discrete-time risk model. The model (1) with N = 1 is a discrete version of the continuous-time Cramér–Lundberg model (also known as the classical risk process) (2) W ˜ u ( t ) = x + c t − ∑ i = 1 P t ξ i , t ⩾ 0 , where, analogously as in model (1), x ⩾ 0 represents initial surplus, c > 0 premium, ξ i are independent and identically distributed nonnegative random variables, and P t is the counting Poisson process with intensity λ > 0. The model (2) can be further extended, cf. E. Spare Andersen’s model [2] or models considered in [6, 7].

Being curious whether initial surplus and collected premiums can always cover incurred claims, for the N-seasonal discrete-time risk model (1) we define the finite time survival probability (3) φ ( u , T ) : = P ( ⋂ n = 1 T { W u ( n ) > 0 } ) = P ( sup 1 ⩽ n ⩽ T ∑ i = 1 n ( X i − κ ) < u ) , where T is some fixed natural number, and the ultimate time survival probability (4) φ ( u ) : = P ( ⋂ n = 1 ∞ { W u ( n ) > 0 } ) = P ( sup n ⩾ 1 ∑ i = 1 n ( X i − κ ) < u ) . Computation of finite time survival (or ruin, ψ ( u , T ) : = 1 − φ ( u , T ) ) probability (3) is far easier than the computation of ultimate time survival (or ruin, ψ ( u ) : = 1 − φ ( u )) probability (4), see Theorem 4 for the precise expressions of φ ( u , T ). Difficulties computing φ ( u ) arise due to φ ( κ N ) , φ ( κ N + 1 ) , … being expressible via φ ( 0 ) , φ ( 1 ) , … , φ ( κ N − 1 ) which are initially unknown, see the formula (5) in the next section. Therefore, the essence of the problem we solve is nothing but finding these initial values φ ( 0 ) , φ ( 1 ) , … , φ ( κ N − 1 ). In this work, we demonstrate that the required values of φ ( 0 ) , φ ( 1 ) , … , φ ( κ N − 1 ) satisfy a certain system of linear equations (see the system (16)) whose coefficients are based on certain polynomials and the roots of s κ N = G S N ( s ), where s ∈ C and G S N ( s ) is the probability-generating function of S N = X 1 + ⋯ + X N .

Let us briefly overview the history and some fundamental works on the subject and mention a few recent papers. The foundation of ruin theory dates back to 1903 when Swedish actuary Filip Lundberg published his work [29], which was republished in 1930 also by Swedish mathematician Harald Cramér, while the random walk formulation, as such, was first introduced by English mathematician and biostatistician Karl Pearson [33]. The Cramér–Lundberg risk model (2) was extended by Danish mathematician Erik Sparre Andersen by allowing claim inter-arrival times to have arbitrary distribution [2]. The next famous works were published in the late eighties by Hans U. Gerber and Elias S. W. Shiu, see [16, 15, 38, 39]. Equally, in the second half of the twentieth century, there were many sound studies regarding the random walk by such authors as William Feller, Frank L. Spitzer, David G. Kendal, Félix Pollaczek and others, see [14, 40, 41, 23, 35] and related works. Scrolling across the timeline in recent decades, one may reference a notable survey [27]. Various assumptions on random walk’s structure in models (1) or (2) (cf. [20, 37]), the variety of other numerical characteristics of renewal risk models than the defined ones in (3) and (4) (cf. [28, 24]), and the research methods (cf. [42]) are the reasons which make the recent literature voluminous. Next to the mentioned references, see [11, 36, 4, 12, 10, 34, 26, 25, 32, 8] as the recent ones on the subject, too.

This section starts by deriving the basic recurrent relation for the ultimate time survival probability. The definition (4), the law of total probability and rearrangements imply (5) φ ( u ) = P ( ⋂ n = 1 ∞ { W u ( n ) > 0 } ) = P ( ⋂ n = 1 N { u + κ n − ∑ i = 1 n X i > 0 } ∩ ⋂ n = N + 1 ∞ { u + κ n − ∑ i = 1 n X i > 0 } ) = P ( ⋂ n = 1 N { ∑ i = 1 n X i ⩽ u + κ n − 1 } ∩ ∩ ⋂ n = N + 1 ∞ { u + κ N − ∑ i = 1 N X i + κ ( n − N ) − ∑ i = N + 1 n X i > 0 } ) = ∑ i 1 ⩽ u + κ − 1 i 1 + i 2 ⩽ u + 2 κ − 1 ⋮ i 1 + i 2 + ⋯ + i N ⩽ u + κ N − 1 P ( X 1 = i 1 ) P ( X 2 = i 2 ) ⋯ P ( X N = i N ) φ ( u + κ N − ∑ j = 1 N i j ) . Substituting u = 0 into the recursive formula (5), we notice that in order to find φ ( κ N ) we need to know the values of φ ( 0 ) , φ ( 1 ) , … , φ ( κ N − 1 ). Moreover, if we know the values of φ ( 0 ) , φ ( 1 ) , … , φ ( κ N − 1 ), the same recurrence (5) allows us to compute φ ( u ) for any u ⩾ κ N by substituting u = 0 , 1 , … there. Thus, as mentioned in the introduction, all we need is a way to compute these initial values.

We now define a series of notations. Recalling that we aim to know the distribution of sup n ⩾ 1 ∑ i = 1 n ( X i − κ ), we define N random variables: (6) M 1 : = sup n ⩾ 1 ( ∑ i = 1 n ( X i − κ ) ) + , M 2 : = sup n ⩾ 2 ( ∑ i = 2 n ( X i − κ ) ) + , … , M N : = sup n ⩾ N ( ∑ i = N n ( X i − κ ) ) + , where x + = max { 0 , x }, x ∈ R, is the positive part function. Obviously, same as X 1 , X 2 , … , X N , every random variable M 1 , M 2 , … , M N attains the values from the set {0, 1, …}.

Let us denote the probability mass functions of M j , their generators X j and the sum S N = X 1 + X 2 + ⋯ + X N : (7) m i ( j ) : = P ( M j = i ) , x i ( j ) : = P ( X j = i ) , s i ( N ) : = P ( S N = i ) , where j ∈ { 1 , 2 , … , N } and i ∈ N 0 . Let F X j ( i ) be the distribution function of the random variable X j , i.e. F X j ( u ) = P ( X j ⩽ u ) = ∑ i = 0 u x i ( j ) , j ∈ { 1 , 2 , … , N } , u ∈ N 0 . In addition, let S ‾ 1 ( 0 ) : = { s ∈ C : | s | ⩽ 1 }, S 1 ( 0 ) : = { s ∈ C : | s | < 1 } be the circles in the complex plane centered at the origin with radius one and denote the probability-generating function of some nonnegative and integer-valued random variable X, G X ( s ) : = ∑ i = 0 ∞ s i P ( X = i ) = E s X , s ∈ S ‾ 1 ( 0 ) .

The definition of the survival probability (4) and the definition of random variable M 1 imply (8) φ ( u + 1 ) = P ( M 1 ⩽ u ) = ∑ i = 0 u m i ( 1 ) for all u ∈ N 0 . Thus, the survival probability computation turns into the setup of the distribution function of M 1 . It is simple to explain the core idea of the paper, i.e. how the probabilities m i ( 1 ) , i ∈ N 0 , are attained. Let us refer to Feller’s book [14, Theorem on page 198]. The referenced Theorem states that if N = 1 in model (1), i.e. the random walk { ∑ i = 1 n X i , n ∈ N } is generated by independent and identically distributed random variables, which are the copies of X, then ( M 1 + X − κ ) + = d M 1 . For arbitrary number of periods N ∈ N the mentioned distribution property is as follows: (9) ( M 1 + X ˜ N − κ ) + = d M N and ( M j + X j − 1 − κ ) + = d M j − 1 , for all j = 2 , 3 , … , N, where X ˜ N is an independnet copy of X N ; see Lemma 2 in Section 5. Metaphorically, the distributions’ equalities in (9) mean that the random variables M 1 , M 2 , … , M N “can see each other”, and, more precisely, based on the equalities in (9), we can set up a system of corresponding equalities of probability-generating functions (10) E s ( M 1 + X ˜ N − κ ) + = E s M N E s ( M 2 + X 1 − κ ) + = E s M 1 ⋮ E s ( M N + X N − 1 − κ ) + = E s M N − 1 . The system (10) contains the desired information on m i ( 1 ) , i ∈ N 0 .

In general, the random variables M 1 , M 2 , … , M N can be extended, i.e. lim u → ∞ P ( M j = u ) > 0, j = 1 , 2 , … , N. However, lim u → ∞ P ( M j < u ) = 1 for all j = 1 , 2 , … , N if E S N < κ N, see Lemma 1 in Section 5.

The condition E S N < κ N is called the net profit condition and it is crucially important for the survival probability φ ( u ). Intuitively, an insurer has no chance to survive in long-term activity, if threatening amounts of claims on average are greater or equal to the collected premiums. This can also be well illustrated by the expected value of W u ( n ) in (1). For instance, E W u ( n N ) = u + n ( − E S N + κ N ) < 0 if E S N > κ N and n is sufficiently large. Consequently, the negative value of W u ( n ) is unavoidable. See Theorem 3 in Section 3 for the precise expressions of the survival probability φ ( u ) , u ∈ N 0 when the net profit condition is violated, i.e. E S N ⩾ κ N.

In this section, based on the previously introduced notations and explanation that our goal is to know the probability mass function of M 1 , we formulate two main theorems for the ultimate time survival probability computation under the net profit condition. Theorem 1, being implied by system (10), provides the relations between m i ( 1 ) , m i ( 2 ) , … , m i ( N ) and x i ( 1 ) , x i ( 2 ) , … , x i ( N ) for all i ∈ N 0 and lays down the foundation for the computation of the ultimate time survival probability φ ( u + 1 ) = ∑ i = 0 u m i ( 1 ) , u ∈ N 0 .

Let us comment on how Theorem 1 is used to obtain the distribution of M 1 , i.e. m i ( 1 ) , i ∈ N 0 . First, we note that the equation s κ N = G S N ( s ) has exactly κ N − 1 roots in s ∈ S ‾ 1 ( 0 ) ∖ { 1 } counted with their multiplicities, see Lemma 4 in Section 5. We denote these roots by α 1 , α 2 , … , α κ N − 1 . We then create the system of linear equations (see eq. (16)) by replicating the equation (13) κ N − 1 times over the roots α 1 , α 2 , … , α κ N − 1 and include (11) as the last equation. To illustrate that explicitly, we define the matrices M 1 , M 2 , … , M N and G 2 , G 3 , … , G N :

In general, it is not easy to give an explicit solution of system (16) or even to prove that the system’s determinant of size κ N × κ N never equals zero. For instance, if N = 1 and κ ∈ N, then the system (16) is (24) ∑ j = 0 κ − 1 α 1 j F X ( j ) ∑ j = 0 κ − 2 α 1 j + 1 F X ( j ) … α 1 κ − 1 x 0 ⋮ ⋮ ⋱ ⋮ ∑ j = 0 κ − 1 α κ − 1 j F X ( j ) ∑ j = 0 κ − 2 α κ − 1 j + 1 F X ( j ) … α κ − 1 κ − 1 x 0 ∑ j = 0 κ − 1 x j ( κ − j ) ∑ j = 0 κ − 2 x j ( κ − j − 1 ) … x 0 m 0 ( 1 ) ⋮ m κ − 2 ( 1 ) m κ − 1 ( 1 ) = 0 ⋮ 0 κ − E X , where X = d X 1 , x i = x i ( 1 ) and α 1 , α 2 , … , α κ − 1 are the simple roots of s κ = G X ( s ) when s ∈ S ‾ 1 ( 0 ) ∖ { 1 }. Then, if x 0 > 0, the determinant of the system’s matrix in (24) is the Vandermonde-like one, x 0 κ ( − 1 ) κ + 1 ∏ j = 1 κ − 1 ( α j − 1 ) ∏ 1 ⩽ i < j ⩽ κ − 1 ( α j − α i ) ≠ 0 , and the probabilities m 0 ( 1 ) , m 1 ( 1 ) , … , m κ − 1 ( 1 ) together with the survival probabilities φ ( 0 ) , φ ( 1 ) , … , φ ( κ ) admit neat closed-form expressions, see [17, Thm. 4]. On the other hand, if κ = 1 and N ∈ N, then the system (16) is (25) x 0 ( N ) x 0 ( 1 ) G X N ( α 1 ) α 1 … x 0 ( N − 1 ) G X N + X 1 + ⋯ + X N − 2 ( α 1 ) α 1 N − 1 x 0 ( N ) x 0 ( 1 ) G X N ( α 2 ) α 2 … x 0 ( N − 1 ) G X N + X 1 + ⋯ + X N − 2 ( α 2 ) α 2 N − 1 ⋮ ⋮ ⋱ ⋮ x 0 ( N ) x 0 ( 1 ) G X N ( α N − 1 ) α N − 1 … x 0 ( N − 1 ) G X N + X 1 + ⋯ + X N − 2 ( α N − 1 ) α N − 1 N − 1 x 0 ( N ) x 0 ( 1 ) … x 0 ( N − 1 ) m 0 ( 1 ) m 0 ( 2 ) ⋮ m 0 ( N − 1 ) m 0 ( N ) = 0 0 ⋮ 0 N − E S N , where α 1 , α 2 , … , α N − 1 are the simple roots of s N = G S N ( s ) in s ∈ S ‾ 1 ( 0 ) ∖ { 1 }, see [19]. If N = 3, the main matrix in (25) is (26) A : = x 0 ( 3 ) x 0 ( 1 ) G X 3 ( α 1 ) α 1 x 0 ( 2 ) G X 3 + X 1 ( α 1 ) α 1 2 x 0 ( 3 ) x 0 ( 1 ) G X 3 ( α 2 ) α 2 x 0 ( 2 ) G X 3 + X 1 ( α 2 ) α 2 2 x 0 ( 3 ) x 0 ( 1 ) x 0 ( 2 ) and one may check that for s 0 ( 3 ) = P ( X 1 + X 2 + X 3 = 0 ) > 0 the matrix A is nonsingular iff ( G X 3 ( α 1 ) α 1 − 1 ) ( G X 3 + X 1 ( α 2 ) α 2 2 − 1 ) ≠ ( G X 3 ( α 2 ) α 2 − 1 ) ( G X 3 + X 1 ( α 1 ) α 1 2 − 1 ) where α 1 , α 2 are the simple roots of s 3 = G X 1 + X 2 + X 3 ( s ) in s ∈ S ‾ 1 ( 0 ) ∖ { 1 }. Computer computations with some chosen random variables X 1 , X 2 , … , X N , N ⩾ 3, do not reveal any examples that the system’s matrix in (16) is singular. The listed thoughts raise the following conjecture.