The insurance model when the amount of claims depends on the state of the insured person (healthy, ill, or dead) and claims are connected in a Markov chain is investigated. The signed compound Poisson approximation is applied to the aggregate claims distribution after

This paper is motivated by the insurance model in which the insured is described by a random variable (rv) with three states (healthy, ill, dead), and rvs are connected in a Markov chain. We assume that the insurer pays one unit of money in the case of illness and continuously pays

The matrix of transition probabilities

It is assumed that at the beginning the insured person is healthy. Hence, the initial distribution is given by

In this paper, we consider triangular arrays of rvs (the scheme of series), i.e. all transition probabilities

All results are obtained under the condition

Here

We denote by

If

In the proofs, we apply the following well-known relations:

The compound Poisson approximation is frequently used to approximate aggregate losses in risk models (see, for example, [

Let

The main part of the approximation

For convenience we present all Fourier transforms of measures used for construction of approximations in a separate table. Note that all measures are denoted by the same capital letters as their Fourier transforms (for example,

The measures can be easily found from their Fourier transforms using the formula

Since

The other measures can be calculated analogously using their Fourier transforms presented in Table

Fourier transforms of used measures.

We analyze the scheme of series, when transition probabilities may differ from one time period to another time period, that is, transition probabilities depend on

Observe that, since

Unlike (

This accuracy is reached, when

Observe that the accuracy of approximation in (

If both probabilities are uniformly separated from zero,

Observe that, if the scheme of sequences is analyzed, all probabilities do not depend on

The local estimates in Theorem

In insurance models, tail probabilities are very important, see, for example [

The non-uniform estimate for distribution functions (

When

We begin from the inversion inequalities.

Observe that (

The characteristic function method is used for the analysis of the model. Therefore our next step is to obtain

The characteristic function

Expression (

We find the eigenvalues by solving the following equation:

Similarly, system (

It is not difficult to notice that

Observe that

Next we prove that

From Lemma

The following estimate shows that

It is not difficult to check that

Let

Notice that

Finally,

Next we demonstrate that

From Lemma

To approximate

The expansion of

The following three lemmas are needed for the approximation of

The proof is very similar to the proof of Lemma

From Corollary

Applying (

Since

It is obvious that

We will use the following simple inequality

By applying Lemma

The second inequality of the lemma is proved similarly. □

From Lemma

Notice that

From Corollary

Using Corollary

By applying (

The second inequality of this lemma is proved similarly. □

Since

By applying (

All inequalities are based on the previously obtained estimates of

Applying inversion formula (

The local estimate is obtained analogously by applying inversion formula (

The proof is similar to the proof of Theorem

Taking into account Corollary

From inversion formula (

From Lemma

By applying inversion formula (

We use the inequalities obtained in the proof of Theorem

Hence,

Summing those inequalities, we get

In order to prove the second inequality of the theorem, we apply the inversion formula (

The summands can be estimated by using the inequalities from the proof of Theorem

Thus, we get

By summing the above inequalities we arrive at