Let me now denote by

S_n renewal process is equal to S_n-1 plus xi_n.

Here, xi_1, xi_2, and so on.

This is a sequence of independent,

identically distributed random variables which almost surely positive distribution.

I would like to formulate two theorems.

The first theorem says as the following.

Assume that the mathematical expectation xi_1,

which I'll denote by mu is finite.

Then N_t counting process which corresponds to this renewal process divided by t,

converges as t goes to infinity to one divided by mu.

And this convergence can be understood in almost sure sense,

that it is a probability of all omegas such that N_t divided

by t converges to one divided by m is equal to one.

Basically, this theorem is an analogue of the law of large numbers or to be more precise,

to the strong law of large numbers.

You know that in the probability theory,

there is one essential fact known as law of large numbers,

which tells us that the xi_1,

xi_2 and so on is a sequence of

independent identically distributed random variables, then,

their sum divided by n converges to the mathematical expectation of xi_1,

that is to mu as n goes to infinity.

And this convergence can be understood in almost surely sense.

So, this theorem is an analogue of the strong law of

large numbers applied for renewal processes.

Okay.

So, theorem two is analogue of the central limit theory.

Let me additionally assume that the second moment is of xi_1 is finite.

And let me denote the variance of xi_1 by sigma squared.

In this case, the theorem yields that

N_t minus t divided by mu divided by sigma square root of t

divided by mu in the power one and a half converges in

distribution to the standard normal law.

Well, this convergence basically means that

if we denote what is written in the left hand side by Z_t,

then the probability that Z_t is less or equals than X

converges as t goes to infinity to

the probability that standard normal random variable is less or equals than X.

This probability is equal to the integral from minus infinity to x,

one divided by square root of 2 pi exponent is of

power minus mu square root divided by two du.

This fact is an analogue of the central limit theorem,

one of the most popular theories from the probability theories.

This result tells us that if xi_1,

xi_2 and so on is a sequence of independent identically distributed random variables,

whose finite second moment is that the sum xi_1 plus so on plus xi_n minus n

multiplied by the mathematical expectation divided by sigma square root of n,

convergence is a weak sense,

the same as distribution,

standard normal random variable.

So, once more, we have here two theorems.

The first one is an analogue of the strong law of large numbers.

The second one is an analogue of the central limit theorem.

Now, I'm going to prove both statements,

and let me start with the first one.

So, to start the proof of theorem one,

let me first mention that the following cross interest inequality holds.

Basically, S at the point capital N of t is less or equal than t and is

less or equal than S at the point N_t+1.

To show that these inequalities hold,

let me just draw a picture and everything will be clear.

So, you know that the plot of renewal process looks as follows.

It jumps at one,

and the moments S_1,

S_2, and so on.

And what we have here is fixed a point t on this plot.

For instance here, that N_t is equal to the y coordinate of this point.