In this paper, the distribution function

Many probabilistic models estimating the likelihoods of certain events are based on the sequence of sums

Implemented using the RandomChoice function in Wolfram Mathematica [

Lines

We say that the fixed natural number

Being curious whether initial surplus and collected premiums can always cover incurred claims, for the

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 [

This section starts by deriving the basic recurrent relation for the ultimate time survival probability. The definition (

We now define a series of notations. Recalling that we aim to know the distribution of

Let us denote the probability mass functions of

The definition of the survival probability (

In general, the random variables

The condition

In this section, based on the previously introduced notations and explanation that our goal is to know the probability mass function of

Let us comment on how Theorem

and set up the system

where ∘ denotes the Hadamard matrix product, also known as the element-wise product, entry-wise product, or Schur product, i.e. two elements in corresponding positions in two matrices are multiplied, see, for example, [

Clearly, the solution of (16) (see Section

The next theorem states that if the net profit condition is unsatisfied, the ultimate time survival is impossible except in some cases when

The last theorem provides an algorithm for the computation of finite time survival probability

The formulated Theorems

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

Let us comment on how the system (16) gets modified if there are multiple roots among

In addition, when some roots of

Once again, computational examples with some chosen random variables

In this section, we formulate and prove several auxiliary lemmas that are later used to prove theorems formulated in Section

We prove the case

According to the strong law of large numbers

It follows that for any such

We prove the equality

Let us demonstrate how the first equality in (

By Lemma

The next lemma provides the quantity and location of the roots of

We follow the proof of [

Thus, there remain

In this section, we prove all of the theorems formulated in Section

We first prove equality (

It is obvious that the right-hand side of (

We now consider the equation (

We now prove equality (

To compute the limit in (

Proof’s direction of (

It remains to prove the equalities in system (

Let us rewrite the system (

We first show that

Let us now consider the case when

Finally, let us consider the case when

The proof of equalities (

In this section, we illustrate the applicability of theorems formulated in Section

We say that a random variable

Let

Let us observe that in the considered example the net profit condition is satisfied

We provide the obtained survival probabilities in Table

The provided values of

Survival probabilities for

Based on Theorem

Let us consider the model (

According to (42) and (43), we check that the net profit condition is satisfied:

Once again, the results obtained in Table

Survival probabilities for

0 | 1 | 2 | 3 | 4 | 5 | 10 | 20 | 30 | 40 | 50 | |

0.048 | 0.127 | 0.209 | 0.286 | 0.355 | 0.417 | 0.649 | 0.873 | 0.954 | 0.983 | 0.994 |

Theorem

Let us consider the bi-seasonal model (

Probability distribution of random variable

Probability distribution of random variable

We find the survival probability

It is easy to observe that

Solving

We then employ (

Then

It follows that