is a mapping from T to R which appears when we fix an element from omega.

So, we have stochastic process Xt and then, we fix omega.

And don't forget we get a function from T to

R. And this is exactly what we can observe in real life.

So, imagine that you analyze a stock price.

This is a curve, actually, from t,

I'm mapping from t to some St. And what you'll know is just a trajectory of this process.

So this is a complete analog of what will happen medical statistics.

We have some samples x1,

x2, and so on.

And we assume that this sample was

obtained as a realization of a set of random variables.

Well, sometimes you have one random variable,

that we assume that it's a random variable,

was observed many times.

So, what you have in real life is a set of real numbers.

Well, what we have here is just a curve which depends on t. Good.

This is a trajectory.

On the other side, there is a notion of finite dimensional distribution.

This is actually a distribution of a random vector Xt1,

Xt2, and so on Xtn.

Where t1, t2 and tn are just some time moments.

These are elements from R. And this is

exactly the main difference

between the probability theory and the theory of stochastic processes.

In the probability theory,

since elements would be independent.

And therefore, the final dimensional distribution is something which is trivial.

You can simply calculate the final dimensional distribution of each component,

and then just say that all of these compounds are independent.

And here, the theory of stochastic processes is essentially dependent.

And therefore, the calculation of

final dimensional distribution is sometimes rather challenging.

Let me provide one rather simple example.

Assumes as a process Xt is equal to xai multiplied by t,

while xai is equal to 1,

with probability one half and is equal to two with probability one half.

So, there is only one source of randomness.

This is as xai.

And this xai determines the process Xt.

What is the trajectory of this process?

Basically, there are two possible outcomes.

The first one when xai equal to one and in this case Xt is equal to

T. And now the outcome is when xai is equal to two.

And in this case, we have a trajectory 2t.

So, there are only two possible trajectories for the process Xt.

As for the final dimensional distributions,

let me calculate for instance the probabilities that Xt1 is

less or equal than X1 and Xt2 is less or equal than x2.

It is an easy exercise to shows that this probability is equal to 0.

If minimum between x1 divided by t1 and X2 divided by t2 is less than 1.

Moreover, it is equal to one half,

if this minimum is from 1 to 2.

And this is equal to 1,

if this minimum is larger than 2, larger or equal than 2.

I would like to leave the exact calculations as a simple exercise.

And basically, this topic is not difficult at all.

From many processes, one can easily plot a trajectory,

at least a typical trajectory of a process on a two-dimensional plot and basically,

the calculation of final dimensional distributions in many cases,

also possible at least for two-dimensional case.