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STOCHASTIC PROCESS - INTRODUCTION

Dr.V. Senthilkumar

PG & Research Department of Mathematics

CPA College ,Bodinayakanur

July 2024

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STOCHASTIC PROCESSES

Stochastic processes are processes that proceed randomly in time.

Rather than consider fixed random variables X, Y, etc. or even

sequences of i.i.d random variables, we consider sequences X₀, X₁,

X₂, …. Where X_t represent some random quantity at time t.

In general, the value X_t might depend on the quantity X_t-₁ at time t-1,

or even the value X_s for other times s < t.

Example: simple random walk .

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STOCHASTIC PROCESS - DEFINITION

A stochastic process is a family of time indexed random variables X_t

where t belongs to an index set. Formal notation, where I is

an index set that is a subset of R.

Examples of index sets:

1) I = (-∞, ∞) or I = [0, ∞]. In this case X_t is a continuous time

stochastic process.

2) I = {0, ±1, ±2, ….} or I = {0, 1, 2, …}. In this case X_t is a discrete

time stochastic process.

We use uppercase letter {X_t } to describe the process. A time series,

{x_t } is a realization or sample function from a certain process.

We use information from a time series to estimate parameters and

properties of process {X_t }.

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PROBABILITY DISTRIBUTION OF A PROCESS

For any stochastic process with index set I, its probability

distribution function is uniquely determined by its finite dimensional

distributions.

The k dimensional distribution function of a process is defined by

for any and any real numbers x₁, …, x_k .

The distribution function tells us everything we need to know about

the process {X_t }.

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MOMENTS OF STOCHASTIC PROCESS

We can describe a stochastic process via its moments, i.e.,

We often use the first two moments.

The mean function of the process is

The variance function of the process is

The covariance function between X_t , X_s is

The correlation function between X_t , X_s is

These moments are often function of time.

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STATIONARY PROCESSES

A process is said to be strictly stationary if has the same

joint distribution as . That is, if

If {X_t } is a strictly stationary process and then, the mean

function is a constant and the variance function is also a constant.

Moreover, for a strictly stationary process with first two moments

finite, the covariance function, and the correlation function depend

only on the time difference s.

A trivial example of a strictly stationary process is a sequence of i.i.d random variables.

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WEAK STATIONARITY

Strict stationarity is too strong of a condition in practice. It is often

difficult assumption to assess based on an observed time series x₁,…,x_k.

In time series analysis we often use a weaker sense of stationarity in

terms of the moments of the process.

A process is said to be nth-order weakly stationary if all its joint

moments up to order n exists and are time invariant, i.e., independent of

time origin.

For example, a second-order weakly stationary process will have

constant mean and variance, with the covariance and the correlation

being functions of the time difference along.

A strictly stationary process with the first two moments finite is also a

second-ordered weakly stationary. But a strictly stationary process may

not have finite moments and therefore may not be weakly stationary.

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THE AUTOCOVARIANCE AND AUTOCORRELATION FUNCTIONS

For a stationary process {X_t }, with constant mean μ and constant

variance σ₂. The covariance between X_t and X_t+s is

The correlation between X_t and X_t+s is

Where

As functions of s, γ(s) is called the autocovariance function and ρ(s)

is called the autocorrelation function (ATF). They represent the

covariance and correlation between X_t and X_t+s from the same process,

separated only by s time lags.

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PROPERTIES OF Γ(S) AND Ρ(S)

For a stationary process, the autocovariance function γ(s) and the autocorrelation function ρ(s) have the following properties:

The autocovariance function γ(s) and the autocorrelation function ρ(s) are positive semidefinite in the sense that

for any real numbers

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CORRELOGRAM

A correlogram is a plot of the autocorrelation function ρ(s) versus the lag s where s = 0,1, ….

Example…

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PARTIAL AUTOCORRELATION FUNCTION

Often we want to investigate the dependency / association between X_t and X_t+k adjusting for their dependency on X_t+₁, X_t+₂,…, X_t+k_-1.

The conditional correlation Corr(X_t , X_t+k | X_t+₁, X_t+₂,…, X_t+k_-1) is usually referred to as the partial correlation in time series analysis.

Partial autocorrelation is usually useful for identifying autoregressive models.

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GAUSSIAN PROCESS

A stochastic process is said to be a normal or Gaussian process if its joint probability distribution is normal.

A Gaussian process is strictly and weakly stationary because the normal distribution is uniquely characterized by its first two moments.

The processes we will discuss are assumed to be Gaussian unless mentioned otherwise.

Like other areas in statistics, most time series results are established for Gaussian processes.

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WHITE NOISE PROCESSES

A process {X_t} is called white noise process if it is a sequence of uncorrelated random variables from a fixed distribution with constant mean μ (usually assume to be 0) and constant variance σ².

A white noise process is stationary with autocovariance and autocorrelation functions given by ….

A white noise process is Gaussian if its joint distribution is normal.

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ESTIMATION OF THE MEAN

Given a single realization {x_t} of a stationary process {X_t}, a natural estimator of the mean is the sample mean

which is the time average of n observations.

It can be shown that the sample mean is unbiased and consistent estimator for μ.

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SAMPLE AUTOCOVARIANCE FUNCTION

Given a single realization {x_t} of a stationary process {X_t}, the sample autocovariance function given by

is an estimate of the autocivariance function.

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SAMPLE AUTOCORRELATION FUNCTION

For a given time series {x_t}, the sample autocorrelation function is given by

The sample autocorrelation function is non-negative definite.

The sample autocovariance and autocorrelation functions have the same properties as the autocovariance and autocorrelation function of the entire process.