# [TESTS] – Equations

$\begin{array}{rl}P\left({X}_{{t}_{1}}\le {x}_{1},{X}_{{t}_{2}}\le {x}_{2},\dots ,{X}_{{t}_{k}}\le {x}_{k}\right)& =F\left({x}_{{t}_{1}},{x}_{{t}_{2}},\dots ,{x}_{{t}_{k}}\right)\\ & =F\left({x}_{h+{t}_{1}},{x}_{h+{t}_{2}},\dots ,{x}_{h+{t}_{k}}\right)\\ & =P\left({X}_{h+t-1}\le {x}_{1},{X}_{h+{t}_{2}}\le {x}_{2},\dots ,{X}_{h+{t}_{k}}\le {x}_{k}\right)\end{array}$ ${X}_{t}$

A stationary process

$\left\{X-t,\phantom{\rule{thinmathspace}{0ex}}t\in \mathbf{\text{N}}\right\}$ 

is said to be strictly or strongly stationary if its statistical distributions remain unchanged after a shift o the time scale. Since the distributions of a stochastic process are defined by the finite-dimensional distribution functions, we can formulate an alternative definition of strict stationarity. If in every n

$n$

, every choice of times t1,t2,,tnN

${t}_{1},{t}_{2},\dots ,{t}_{n}\in \mathbf{\text{N}}$

and every time lag k

$k$

such that ti+kN

${t}_{i+k}\in \mathbf{\text{N}}$

, the n

$n$

-dimensional random vector (Xt1+k,Xt2+k,,Xtn+k)

$\left({X}_{{t}_{1}+k},{X}_{{t}_{2}+k},\dots ,{X}_{{t}_{n}+k}\right)$

has the same distribution as the vector (Xt1,Xt2,,Xtn)

$\left({X}_{{t}_{1}},{X}_{{t}_{2}},\dots ,{X}_{{t}_{n}}\right)$

, then the process is strictly stationary. That is for h

$h$

and xi

${x}_{i}$

P(Xt1x1,Xt2x2,,Xtkxk)=F(xt1,xt2,,xtk)=F(xh+t1,xh+t2,,xh+tk)=P(Xh+t1x1,Xh+t2x2,,Xh+tkxk)

$\begin{array}{rl}P\left({X}_{{t}_{1}}\le {x}_{1},{X}_{{t}_{2}}\le {x}_{2},\dots ,{X}_{{t}_{k}}\le {x}_{k}\right)& =F\left({x}_{{t}_{1}},{x}_{{t}_{2}},\dots ,{x}_{{t}_{k}}\right)\\ & =F\left({x}_{h+{t}_{1}},{x}_{h+{t}_{2}},\dots ,{x}_{h+{t}_{k}}\right)\\ & =P\left({X}_{h+t-1}\le {x}_{1},{X}_{h+{t}_{2}}\le {x}_{2},\dots ,{X}_{h+{t}_{k}}\le {x}_{k}\right)\end{array}$

for any time shift h

$h$

and observation xj

${x}_{j}$

. If {Xt,tN}

$\left\{X-t,\phantom{\rule{thinmathspace}{0ex}}t\in \mathbf{\text{N}}\right\}$

is strictly stationary, then the marginal distribution of Xt

${X}_{t}$

is independent of t

$t$

. Also,the two-dimensional distributions of (Xt1,Xt2)

$\left({X}_{{t}_{1}},{X}_{{t}_{2}}\right)$

are independent of the absolute location of t1

${t}_{1}$

and t2

${t}_{2}$

, only the distance t1t2

${t}_{1}-{t}_{2}$

matters. As a consequence, the mean function E(X)

$E\left(X\right)$

is constant, and the covariance Cov(Xt,Xtk)

$Cov\left({X}_{t},{X}_{t-k}\right)$

is a function of k

$k$

only, not of the absolute location of k

$k$

and t

$t$

. At higher order moments, like the third order moment, E[XuXtXv]

$E\left[{X}_{u}{X}_{t}{X}_{v}\right]$

remains unchanged if one add a constant time shift to s,t,u

$s,t,u$