2 Stochastic Process

Are random process.
Stochastic process is a family of time indexed variables X(w,t) where w belongs to a sample space and t belongs to an indexed set. For a fixed t X(w,t) is a random variable. For a given w, X(w,t) as function of t is called a sample function or a realization. The population of that consists of all possible realization is called ensemble in stochastic process and time series analysis. Thus a time series is a realization of a sample space of a stochastic process.
Consider a finite set of random variables \(X_{t_1},X_{t_2},........,X_{t_n}\). The n- dimensional distribution is given by F(\(X_{t_1},........,X_{t_n}\)).
A simpler way of describing a stochastic process is to give the moments of the first and second moments. i.e mean and variance.

Mean \(f^n: \mu (t)= E(X_t)\)

Variance \(f^n\) \begin {align*} var(X_t)&= E(X_t-\mu (t))^2\\ &=\sigma ^2(t) \end {align*}

The auto co variance function is another, can be used to describe stochastic process defined \[\varphi (t_1,t_2)= E[[X_{t_1}-\mu (t_1)][X_{t_2}-\mu (t_2)]]\]

A process is said to be first order stationary in the distribution in time if its one time variable i.e \(F(X_{t_1})= F(X_{t_{1}+k})\) for any integers \( t_1,t_{1}+k\) and second order stationary in distribution if \(F(X_{t_1},X_{t_2})= F(X_{t_{1}+k},X_{t_{2}+k})\) for the integers \(t_1, t_2, t_{1}+k, t_{2}+k\)

and an n- order stationary if
\(F(X_{t_1},X_{t_2},.....,X_{t_n})= F(X_{t_{1}+k},X_{t_{2}+k},.....,X_{t_{n}+k})\) for integers \(t_1, t_2,......, t_n,t_{1}+k,......., t_{n}+k\)

A process is said to be strictly stationary if \(F(X_{t_1},X_{t_2},.....,X_{t_n})= F(X_{t_{1}+k},.......,X_{t_{n}+k})\) for integers \(t_1, t_2,.......,t_n, k\) and it is true for \(n= 1,2,......\)
in other words s.t means that the joint distribution of \(X_{t_1},X_{t_2},......,X_{t_n}\) is the same as the joint distribution of \(X_{t_1+k},X_{t_{2}+k},..........,X_{t_{n}+k}\) for every \(t_1, t_2,........., t_n, k\). This means shifting the origin by the amount k has no effect on the distribution.
In particular if \(n=1\) it implies that the distribution of \(X(t)\) must be the same \(\forall t\) so that \(\mu (t)= \mu \) and \(\sigma ^2(t)= \sigma ^2\) are constants and do not depend on t.

If \(n= 2\), the joint distribution of \(X_{t_1}\) and \(X_{t_2}\) depends only on \(t_1, t_2\) which is a lag and \(\varphi (t_1,t_2)\) depends on \(t_1-t_2\) and we write \begin {align*} \varphi (k)&= E[[X_t-\mu ][x_{t+k}-\mu ]]\\ &= Cov(X_t,X_{t+k}) \end {align*}

and this is the co variance at lag k.
The auto correlation \(f^n\) can now be given as.

which measures the correlation between \(X_t\) and \(X_{t+k}\).

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