Week 5.3: Spectral density of a wide-sense stationary process-1

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來自 National Research University Higher School of Economics 的課程

Stochastic processes

29 個評分

National Research University Higher School of Economics

Stochastic processes

29 個評分

The purpose of this course is to equip students with theoretical knowledge and practical skills, which are necessary for the analysis of stochastic dynamical systems in economics, engineering and other fields. More precisely, the objectives are 1. study of the basic concepts of the theory of stochastic processes; 2. introduction of the most important types of stochastic processes; 3. study of various properties and characteristics of processes; 4. study of the methods for describing and analyzing complex stochastic models. Practical skills, acquired during the study process: 1. understanding the most important types of stochastic processes (Poisson, Markov, Gaussian, Wiener processes and others) and ability of finding the most appropriate process for modelling in particular situations arising in economics, engineering and other fields; 2. understanding the notions of ergodicity, stationarity, stochastic integration; application of these terms in context of financial mathematics; It is assumed that the students are familiar with the basics of probability theory. Knowledge of the basics of mathematical statistics is not required, but it simplifies the understanding of this course. The course provides a necessary theoretical basis for studying other courses in stochastics, such as financial mathematics, quantitative finance, stochastic modeling and the theory of jump - type processes.

從本節課中

Week 5: Stationarity and Linear filters

Upon completing this week, the learner will be able to determine whether a given stochastic process is stationary and ergodic; determine whether a given stochastic process has a continuous modification; calculate the spectral density of a given wide-sense stationary process and apply spectral functions to the analysis of linear filters.

Week 5.1: Two types of stationarity-119:50

Week 5.2: Two types of stationarity-28:59

Week 5.3: Spectral density of a wide-sense stationary process-17:49

Week 5.4: Spectral density of a wide-sense stationary process-24:58

Week 5.5: Stochastic integration of the simplest type10:11

Week 5.6: Moving-average filters-15:59

Week 5.7: Moving-average filters-212:03

Week 5.8: Moving-average filters-38:30

與講師見面

Vladimir Panov
Assistant Professor
Faculty of economic sciences, HSE

[MUSIC]

Now I would like to introduce the notion of spectral density.

There is a very important fact in the probability theory,

so-called Bochner–Khinchin's theorem,

Which states that the function phi,

Is a characteristic function of some random variable psi if and

only if the following three conditions are fulfilled.

First of all, the function phi is continuous,

secondly, it is positive semi-definite.

Since phi is a function from R to the set of complex numbers,

positive semi-definite here means that

the sum zeta j, zeta k complex conjugate.

Multiplied by phi is a point uj minus uk,

should be no negative for

any set of complex numbers Z1 and

so on Zn, And

for an real u1 and so on un.

Here is the sum jk from 1 to n.

And the third property is that the function phi at 0 should be equal to 1.

This is not a difficult exercise to show that if functions' characteristics and

those properties are fulfilled, all of them follow by simple calculations.

Most sophisticated part of this theorem is in your statement.

So it turns out that for any phi which satisfies these assumptions,

there exists some random variable.

Such as that phi(u) is equal to the mathematical expectation

exponent to the power iu psi.

Bochner–Khinchin's theorem is a general mathematical concept,

and it appears in some forms in other fields of mathematics.

Like functional analysis, in real analysis, and other parts in calculus.

And actually there are many alternative formulations of this statement.

For instance, if [INAUDIBLE] to the third property, so

we assume only the first and the second.

Then analog of Bochner–Khinchin's theorem still holds,

namely there exists a measure mu sized at

phi of u is equal to the integral exponent iux multiplied by mu dx.

So mu can be not a probabilistic measure, definitely it is not a probabilistic

measure size phi(0) is not equal to 1, but there is still some measure.

Another alternative formulation is when we assume 1), 2),

and also assume that integral absolute value phi(u), du is finite.

Then in this case the measure mu has a density,

in the sense that the function phi(u) is equal to

the integral exponent iux, s(x)dx.

Those are various alternative formulations, and

in our case it will be more important the last one.

Well, which function we can take is the function phi.

So assume now that we have a process xt

which is weakly stationary.

Therefore, the resistive function gamma out of covariance function,

such that covariance function of the process xt at time points t and

s is equal to gamma as a time point t- s.

If this function gamma is such that

it is continuous, And

also integral gamma (u,du) is finite.

Then we can apply the theorem of Bochner–Khinchin in this form,

because any function gamma is positive semi-definite,

it is a simple exercise to show this.

And actually, this fact follows from the properties of the covariance function.

Therefore, in this case there exists

some function g(x) such that

gamma(u) is equal to the integral

exponent in the power iux g(x) dx.

This basically means that gamma(u)

is a Fourier transform of the function g at the point u.

On the other side, f will get transformed as inverse operation.

So we can say that the function

g(x) is equal to 1 divided by 2 pi integral,

exponent to the power -iux gamma(u)du.

And this function g is actually exactly the spectral

density of a weakly stationary process xt.

This is a definition and there is a discrete analog of this function.

So you know that there is an analog of these statements that are based on

the so-called theorem.

And if we proceed exactly in the same way for discrete time processes,

we'll finally get that the function g = 1 divided by 2 pi.

Sum of all h from minus

infinity to plus infinity,

exponent in the power -iu,

ihx gamma(h).

And in the discrete time,

the spectral density is defined exactly via this formula.

In the next subsection I will provide some examples of the spectral density.

And also I would like to discuss the main properties of this object.

[NOISE]

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