A time series is integrated of order d if

${\displaystyle (1-L)^{d}X_{t}\ }$ 

is a stationary process, where ${\displaystyle L}$  is the lag operator and ${\displaystyle 1-L}$  is the first difference, i.e.

${\displaystyle (1-L)X_{t}=X_{t}-X_{t-1}=\Delta X.}$ 

In other words, a process is integrated to order d if taking repeated differences d times yields a stationary process.

In particular, if a series is integrated of order 0, then ${\displaystyle (1-L)^{0}X_{t}=X_{t}}$  is stationary.

An I(d) process can be constructed by summing an I(d − 1) process:

• Suppose ${\displaystyle X_{t}}$  is I(d − 1)
• Now construct a series ${\displaystyle Z_{t}=\sum _{k=0}^{t}X_{k}}$ 
• Show that Z is I(d) by observing its first-differences are I(d − 1):

${\displaystyle \Delta Z_{t}=X_{t},}$ 

where

${\displaystyle X_{t}\sim I(d-1).\,}$ 

• ARIMA
• ARMA
• Random walk
• Unit root test

• Hamilton, James D. (1994) Time Series Analysis. Princeton University Press. p. 437. ISBN 0-691-04289-6.