# Definition:Successive Values of Time Series

## Definition

Let $T$ be a time series whose observations are $\map z {\tau_1}, \map z {\tau_2}, \dotsb, \map z {\tau_t}, \dotsb$ at an unbroken sequence of timestamps $\tau_1, \tau_2, \dotsb, \tau_t, \dotsb$.

When we have $N$ such successive observations, without a missing value, we write:

$z_1, z_2, \dotsb, z_t, \dotsb, z_N$

and it is understood that the subscripts correspond directly to the timestamps.

### Equispaced Time Series

Let $T$ be equispaced with time interval $h$ between adjacent observations.

Then $N$ successive observations, written as:

$z_1, z_2, \dotsb, z_t, \dotsb, z_N$

occur at timestamps:

$\tau_0 + h, \tau_0 + 2 h, \dotsb, \tau_0 + t h, \dotsb, \tau_0 + N h$

Hence we can refer to the observations at timestamp $\tau_0 + t h$ as $z_t$.

## Sources

Part $\text {I}$: Stochastic Models and their Forecasting:
$2$: Autocorrelation Function and Spectrum of Stationary Processes:
$2.1$ Autocorrelation Properties of Stationary Models:
$2.1.1$ Time Series and Stochastic Processes: Time series