Second property is that FN Star(x) is large or equals than F_n plus 1 Star(x).

This is also a very simple property and it follows

from the fact the event Xi_1 plus so on plus

Xi_n is less or equals than X includes the event that

Xi_1 plus so on plus Xi_n plus 1 is less or equals an X.

This is once more because all sides are non-negative.

And from here, we immediately conclude there's this unequality holds.

Well, this is basically all that you should know about this operation.

There are also a couple of properties like committed to

return the assets activity of the convolution,

I guess it is also quite trivial.

Now, let me show how you can apply these properties in the context offering UL theorem.

The following theorem holds.

Let Sn be a renewal process.

These Sn equals to Sn minus 1 plus Xi_n, Xi_1,

Xi_2 and so on is a sequence of

independent and integral distributed random variables with

the distribution function F and we assume that

these random variables are almost surely positive so this F is equals to zero at zero.

Okay. The theorem has two parts.

The first part would claim that the serious UT equals to

the sum N from one to infinity Fn Star(t).

This sum series converge.

And second item stances that for renewal process Sn the mathematical expectation of

its counting process N_t is equal to exactly N U(t).

So both of the statements are quite important.

So, the first statement.

We have series converge.

And the second that this series give us actually the mathematical expectation of N_t.

Well, the first item can be

proving by applying the properties which I discussed previously.

This is a very technical result and I would like to skip it.

For the second part,

let me prove this statement.

The proof is quite simple.

We just show to apply the definition of the counting process.

That mathematical expectation of t is

a mathematical expectation of the maximum index of N such that S_n is less and equal N_t.

But S_n itself is increasing process at least not decreasing.

Therefore, this is exactly the same as to

calculate the mathematical expectation of the amount of

N such that S_n is less or equals than t. Well,

this equal to the mathematical expectation of

the sum N from one to infinity indicator that

S_n is less or equals than T. Here you can put

the sum outside the mathematical expectation and what you will have here,

is a mathematical expectation of the indicator.

This is nothing more than a probability.

So it's sum from one to infinity probability is that S_n is less or equals N_t.

But the distribution function of S_n is exactly FN Star.

And so, what you have here is this sum FN Star(t).

Okay, we have proven this fact.

And actually it is a very nice result which

gives us a precise formula for the mathematical expectation of N_t.

There is only one very important issue related to this fact,

is that it's almost impossible to calculate the sum directly.

In fact, it should calculate not only the convolution

of N distribution function but you also should find the limit of this series.

And it turns out to be completely impossible task if you have

a distribution of F in general form.

And to solve this problem,

to calculate mathematical expectation from F,

we shall somehow apply this result but non direct way.

In the next subsection,

I would like to show you one approach.

How we can reach this goal,

how we can estimate mathematical expectation of N_t from F.