Distribution of shifted discrete random walk generated by distinct random variables and applications in ruin theory

In this paper, we set up the distribution function $$ \varphi(u)=\mathbb{P}\left(\sup_{n\geqslant 1}\sum_{i=1}^{n}\left(X_i-\kappa\right)<u\right), $$ and the generating function of $\varphi(u+1)$, where $u\in\mathbb{N}_0$, $\kappa\in\mathbb{N}$, the random walk $\left\{\sum_{i=1}^{n}X_i, n\in\mathbb{N}\right\},$ consists of $N\in\mathbb{N}$ periodically occurring distributions, and the integer-valued and non-negative random variables $X_1,\,X_2,\,\ldots$ are independent. This research generalizes two recent works where $\{\kappa=1,\,N\in\mathbb{N}\}$ and $\{\kappa\in\mathbb{N},\,N=1\}$ were considered respectively. The provided sequence of sums $\left\{\sum_{i=1}^{n}\left(X_i-\kappa\right),\,n\in\mathbb{N}\right\}$ generates so-called multi-seasonal discrete-time risk model with arbitrary natural premium and its known distribution enables to calculate the ultimate time ruin probability $1-\varphi(u)$ or survival probability $\varphi(u)$. Verifying obtained theoretical statements we demonstrate several computational examples for survival probability $\varphi(u)$ and its generating function when $\{\kappa=2,\,N=2\}$, $\{\kappa=3,\,N=2\}$, $\{\kappa=5,\,N=10\}$ and $X_i$ admits Poisson and some other distributions. We also conjecture the non-singularity of certain matrices.


Introduction
Many probabilistic models, estimating likelihoods of certain events, are based on the sequence of sums n i=1 X i , n ∈ N , where X i are some random variables.This sequence is called the random walk (r.w.).Random walks are usually visualized as branching trees or certain paths in plane or space; their occurrence spreads from pure mathematics to many applied sciences.For instance, one may refer to Case-Shiller home pricing index [1] or even more generally to the random walk hypothesis [2].From a pure mathematics standpoint, one may mention the random matrix theory, see for example [3], [4], [5] and other related works.Perhaps the closest context where the need to know the distribution of sup n 1 n i=1 (X i − κ) arises is insurance mathematics.In ruin theory one may assume that insurer's wealth W(n) in discrete time moments n ∈ N consists of incoming cash premiums and outgoing payoffs (claims), and W(n) admits the following representation: where u ∈ N 0 := N ∪ {0} is interpreted as initial surplus W(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, non-negative and integer-valued but not necessarily identically distributed.More precisely, 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 + κn and the sum of random variables n i=1 X i when n varies, see Figure 1 3 .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 more general continuous time Cramér -Lundberg model (also known as classical risk process) where, analogously as in model ( 1), x 0 represents initial surplus, c > 0 premium, ξ i independent and identically distributed non-negative 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 [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 where T is some fixed natural number, and the ultimate time survival probability Calculation of finite time survival (or ruin, ψ(u, T ) := 1 − ϕ(u, T )) probability ( 3) is far easier than the calculation of ultimate time survival (or ruin, ψ(u) = 1 − ϕ(u)) probability (4), see Theorem 4 below for ϕ(u, T ) expressions.Difficulties calculating ϕ(u) arise due to ϕ(κN), ϕ(κN+1), . . .being expressible via ϕ(0), . . ., ϕ(κN − 1) which are initially unknowns, 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(κN − 1).In this work, we demonstrate that the required values of ϕ(0), . . ., ϕ(κN − 1) satisfy a certain system of linear equations (see the system (16)), coefficients of which are based on certain polynomials and the roots of s κN = G S N (s), where s ∈ C and G S N (s) is the probabilitygenerating function of S N = X 1 + . . .+ X N .Let us briefly overview the history and some classical 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 [8], 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 [9].Cramér-Lundberg riks model (2) was extended by Danish mathematician Erik Sparre Andersen by allowing claim inter-arrival times to have arbitrary distribution [7].The next famous works were published in the late eighties by Hans U. Gerber and Elias S. W. Shiu, see [10], [11], [12], [13].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], [15], [16], [17], [18] and related works.Scrolling across the timeline in recent decades, one may reference a notable survey [19].Various assumptions on random walk's structure in models (1) or (2) make recent literature voluminous.With that said, let us mention [20], [21], [22], [23], [24], [25], [26], [27], [28], [29].

Recursive nature of ultimate time survival probability, basic notations and the net profit condition
This section starts with deriving the basic recurrent relation for the ultimate time survival probability.The definition (4), the law of total probability and rearrangements imply Substituting u = 0 into the recursive formula (5), we notice that in order to find ϕ(κN) we need to know the values of ϕ(0), ϕ(1), . . ., ϕ(κN − 1).Moreover, if we know the values of ϕ(0), ϕ(1), . . ., ϕ(κN − 1), the same recurrence (5) allows us to calculate ϕ(u) for any u κN by substituting u = 0, 1, . . .there.Thus, as mentioned in the introduction, we only need a way to calculate these initial values.
We now define a series of notations.Recalling that we aim to know the distribution of sup n 1 n i=1 (X i − κ), we define N random variables: 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 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.
Also, for the complex number s ∈ C let us denote the probability-generating function of some non-negative and integer-valued random variable X The definition of the survival probability (4) and the definition of random variable M 1 imply m (1)  i for all u ∈ N 0 .
Thus, the survival probability calculation 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 (1)  i , 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 { n i=1 X i , n ∈ N} is generated by independent and identically distributed random variables, which are the copies of X, then (M 1 + X − κ) For arbitrary number of periods N ∈ N the mentioned distribution property is as follows: see Lemma 2 in Section 5 below.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 The system (10) contains the desired information on m (1)  i , 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 ES N < κN, see Lemma 1 in Section 5 below.
The condition ES N < κN 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(n) in (1).For instance, κN and n is sufficiently large.Consequently, the negative value of W(n) is unavoidable.See Theorem 3 in Section 3 below for precise statements on the survival probability ϕ(u), u ∈ N 0 when the net profit condition is violated, i. e. ES N κN.

Main results
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 ultimate time survival probability calculation under the net profit condition.Theorem 1, implied by system (10), provides the relations between m (1)  i , m (2)  i , . . ., m (N) i and x (1)  i , x (2)  i , . . ., x (N) i for all i ∈ N 0 and lays down the foundation for ultimate time survival probability ϕ(u+1) = u i=0 m (1)  i , u ∈ N 0 calculation.
Theorem 1. Suppose that the N-seasonal discrete-time risk model (1) is generated by random variables X 1 , X 2 , . . ., X N and the net profit condition ES N < κN is satisfied.Then the following statements are correct: 1.The probability mass functions of random variables M 1 , M 2 , . . ., M N and X 1 , X 2 , . . ., X N are related by x (2)  j (s κ − s i+ j ) + . . .
The next theorem states that if the net profit condition is unsatisfied, then ultimate time survival is impossible except in some cases when S N is degenerate.Theorem 3. Suppose that N-seasonal discrete-time risk model (1) is generated by random variables X 1 , X 2 , . . ., X N and the net profit condition is not satisfied.In this case: here n * is equal to such n ∈ {1, 2, . . ., N} with which κn − n k=1 X k attains its minimum.
Theorem 4. For the finite time survival probability (3) of the N-seasonal discrete time risk model defined in (1), holds: and Moreover, for the defined probabilities (19), it holds that The formulated Theorems 1, 2, 3 and 4 are proved in Section 6 below.
where α 1 , α 2 , . . ., α N−1 1 are the simple roots of s N = G S N (s) in |s| 1, see [32].If N = 3, the main matrix in (25) is and one may check that for s (3) Computer calculations 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.
Also, when some roots of s κN = G S N (s) are of multiplicity r ∈ {2, 3, . . ., κN − 1}, then, to avoid identical lines in ( 16), we must replace the corresponding lines by derivatives as provided in equality (14).
Once again, computational examples with some chosen random variables X 1 , X 2 , . . ., X N , N 3 and κ 1 do not reveal any examples that such a modified (due to multiple roots and/or S N not attaining "small" values) system's matrix in ( 16) is singular.
Proof.We prove the case j = 1 only and note that the other cases can be proven similarly.
According to the law of large numbers, almost surely Therefore, Consequently, for any arbitrarily small ε > 0, there exists such number N ε ∈ N that It follows that for any such ε and any u ∈ N we have The last inequality implies where ε > 0 is as small as we want and the assertion of Lemma follows.
Lemma 2. If the net profit condition is satisfied, i. e. ES N < κN, it holds that (M j + X j−1 − κ) + d = M j−1 , for all j = 2, 3, . . ., N, and Proof.We prove the equality (M 1 + X N − κ) + d = M N only and note that the other ones can be proved by the same arguments.According to Lemma 1, the random variable M 1 is not extended, i. e. it does not attain infinity.Let us denote Xj = X j − κ for all j ∈ {1, 2, . . ., N}.Then Lemma 3. Let s ∈ C be the complex number and 0 < |s| 1.Then the probability-generating functions of X 1 , X 2 , . . ., X N and M 1 , M 2 . . ., M N are related the following way Proof.Let us demonstrate how the first equality in ( 27) is derived and note that the remaining ones follow the same logic.By Lemma 2, the distributions' equality (M 1 + X N − κ) + d = M N implies the equality of probability-generating functions G M N (s) = G (M 1 +X N −κ) + (s).Then, applying the law of total expectation for the last equality, we obtain Multiplying both sides of the last equality by s κ when s 0 and observing that we get the desired result.
The next lemma provides the quantity and location of the roots of s κN = G S N (s).Thus, there remain κN − 1 roots of s κN = G S N (s) in |s| 1, s 1 and additionally one can say that |s| = 1 , s 1 is the root of s κN = G S N (s) if the greatest common divisor of κN and the powers of s in G S N (s) is greater than one.

Proofs
In this section, we provide the proofs for four theorems formulated in Section 3.
Proof of Theorem 1.We first prove equality (12).To derive (12), we use the system of equations ( 27) from Lemma 3.According to the conditions of Lemma 3, s 0 and we rearrange the system (27) by multiplying its first equality by 1, the second one by G X N (s)/s κ , the third one by G X N +X 1 (s)/s 2κ and so on till the last equality which we multiply by G X N +X 1 +...+X N−2 (s)/s κ(N−1) .We then add up all these equations and obtain Here we have used the fact that if any random variables X are Y independent, then Thus, the equality ( 12) is proved.We now derive (13).
It is obvious that the right-hand-side of ( 28) equals zero if we set s = α, where α 1 is the root of G S N (s) = s κN , |s| 1.We then divide both sides of (28) by α − 1, i. e.
and get which is the claimed equality (13).
We continue the proof by letting s → 1 − in (29).Because the net profit condition ES N < κN holds and the random variable Calculating the limit in (30) there are two separate cases to examine: EM N < ∞ and EM N = ∞.If EM N < ∞, then the limit in (30) is zero.However, this limit is zero even if EM N = ∞.Indeed, if EM N = ∞, then by using L'Hospital's rule we get see [32,Lem. 5.5]4 .Thus, the limit in ( 30) is zero and the equality (11) in Theorem 1 follows.It remains to prove the equalities in system (15).In short, every equality in system (15) is the corresponding equality from (27) expanded at s = 0. Let us demonstrate the derivation of the first equality in (15) in detail and note that the remaining ones are derived analogously.We need to show that the first equality in ( 27) implies (the first one in (15)) Equality ( 33) is implied by (32) because of the following equalities: when n = κ, κ + 1, . . .The proof of Theorem 1 is finished.
Proof of Theorem 2. Let us rewrite the system (27) the following way 34) and denote this system by AB = C. Determinant of the main matrix in (34) is Thus, the main matrix in (34) is invertible for all such s that s κN G S N (s) and B = A −1 C. Therefore, the previous thoughts and equality (18) imply where M 11 , M 21 , . . ., M N1 are the minors of A and C is the column vector of the right-hand-side of (34).
Proof of Theorem 3. We first show that ES N > κN implies ϕ(u) = 0 for all u ∈ N 0 .The recurrence (5) yields where µ i (u) for each i ∈ {1, 2, . . ., κ(N − 1)} are coefficients consisting of the products of probability mass functions of random variables X 1 , X 2 , . . ., X N .For instance, if N = 2 and κ = 1, then x (1)  i 1 x (2) If µ 0 (u) := s (N) u+κN and µ j (u) := 0 when j > κ(N − 1), then the equality in (35) is By summing up both sides of the last equality by u, which varies from 0 to some natural and sufficiently large v, we obtain We now change the order of summation in (36 and obtain Subtracting v+κN i=0 ϕ(i) v u=i−κN s (N) u+κN−i from both sides of the last equation and rearranging, we get Clearly, the definition of the survival probability (4) implies that ϕ(u) is a non-decreasing function, i. e. ϕ(u) ϕ(u + 1) for all u ∈ N 0 .Thus, there exists a non-negative limit ϕ(∞) := lim u→∞ ϕ(u) and ϕ(∞) = 1 if the net profit condition ES N < κN holds, see Lemma 1.We now let v → ∞ in both sides of (37).For the first sum in (37) we obtain and for the second Indeed, let us recall that EX = ∞ i=0 P(X > i), when X is some non-negative and integer-valued random variable.Then, the upper bound of (39) is while the lower bound is the same due to inequality where M is some fixed and sufficiently large natural number.Thus, when v → ∞, the equality in (37) is If ES N > κN, as under the case of consideration, the non-negative right-hand-side of (40) implies ϕ(∞) = 0 and consequently ϕ(u) = 0 for all u ∈ N 0 .The first case that ES N > κN makes survival impossible is proved.

Numerical examples
In this section, we illustrate the applicability of theorems formulated in Section 3.All of the necessary calculations are performed using Wolfram Mathematica [6].Notice that some of the examples considered here are also considered in [35,Sec. 4], where the ultimate time survival probability was obtained by calculating the limits of certain recurrent sequences.Therefore, in some examples here we check if the obtained values of ϕ(u) match the previously known ones.
We say that a random variable X is distributed according to the displaced Poisson distribution P(λ, ξ) with parameters λ > 0 and ξ ∈ N 0 , if One can check that the following facts for the displaced Poisson distribution are true: 2. If X ∼ P(λ 1 , ξ 1 ), Y ∼ P(λ 2 , ξ 2 ) and also X and Y are independent, then 3. G X (s) = s ξ e λ(s−1) .
Let us recall that the same recurrence (5) can be used to calculate ϕ(u) when u 5.
We provide the obtained survival probabilities in Table 1.The provided values of ϕ(u) match the ones given in [35,Tab. 1], where they were obtained by a different method.
Substituting n = 2, 3, . . .into the last two equations, we obtain m (1)  1 , m (1) 2 , . . .The survival probability ϕ(0) is found using recurrence (5): After completing all the necessary calculations, we get survival probabilities which are provided in Table 2. Once again, obtained results in Table 2 match the ones presented in [35,Tab. 3], where the numbers are obtained differently, i. e. calculating limits of certain recurrent sequences.
Example 3. Let us consider the bi-seasonal model (1) with κ = 3 where claims are represented by two independent random variables X 1 and X 2 , whose distributions are given in Table 3  We find the survival probability ϕ(u) for all u ∈ N 0 and its generating function Ξ(s).
Note that the complex roots always occur in conjugate pairs due to G S N (s) − s κN = G S N (s) − s κN , where over-line denotes conjugate.According to Lemma 4, there must be one root of multiplicity two and one may check that α 1 is such.
We then employ (14) to create the modified versions of M 1 , M 2 and G 2 .Let M1 , M2 and G2 be The correctness of these results can be verified in the following way.If initial surplus u = 1, ruin can only occur at the first moment of time and only if 1 + 3 • 1 − X 1 0, i. e. X 1 = 4. Thus, ϕ(1) = 1 − P(X 1 = 4) = 1 − 0.0016 = 0.9984.If initial surplus u 1, then ruin will never occur.There are two reasons for that.First of all, in the first moment of time insurer's wealth will never drop below one.Moreover, every two periods insurer earns 6 units of currency and that is the maximum amount of claims that the insurer can suffer during two consecutive periods.The result of ϕ(0) is also logical as with no initial capital ruin can occur only if X 1 = 3 or X 1 = 4, thus ϕ(0) = 1 − P(X 1 = 3) − P(X 1 = 4) = 1 − 0.0256 − 0.0016 = 0.9728.
The generating function of ϕ(1), ϕ(2), . . . in the considered case is simple One may verify that Theorem 2 produces the same result.
Example 4. In the last example, we consider ten season model with a premium rate of 5, i. e. N = 10, κ = 5, and we assume claims to be generated by independent random variables X k ∼ P(k/(k + 1) + 4, 0), k ∈ {1, 2, . . ., 10}, where P(λ, 0) denotes Poisson distribution with parameter λ.We calculate both the finite time survival probability ϕ(u, T ) and the ultimate time survival probability ϕ(u) and provide a frame of ultimate time survival probability-generating function Ξ(s).

Lemma 4 .
Assume that the net profit condition G S N (1) = ES N < κN is valid.Then there are exactly κN − 1 roots, counted with their multiplicities, of s κN = G S N (s) in |s| 1, s 1.Proof.We follow the proof of[33, Lemma 9].Due to the estimate|G S N (s)| 1 < λ|s| κNon |s| = 1 when λ > 1, Rouché's theorem implies that both functions G S N (s) − λs κN and λs κN have the same number of roots in |s| < 1 and this number is κN due to the fundamental theorem of algebra.When λ → 1 + some roots of G S N (s) − λs κN remain in |s| < 1 and some migrate to the boundary points |s| = 1.Obviously, s = 1 is always the root of s κN = G S N (s) and it is the simple root because the net profit condition holds, i. e. G S N (s) − s κN s=1 = ES N − κN < 0.

Table 3 :
Probability distribution of random variable X 1