# Definition:Summation Operator

## Definition

Let $f: \R \to \R$ be a real function.

The summation operator $S$ on $f$ is defined as:

$\map S f = \ds \sum_{j \mathop = 0}^\infty \map f {x - j} = \map f x + \map f {x - 1} + \map f {x - 2} + \dotsb$

## Also presented as

The summation operator, when applied to a time series, can be written in terms of the backward shift operator as:

$\map S f = \map {\nabla^{-1} } f = \paren {1 - B}^{-1}$

and so:

$\map S {z_t} = \ds \sum_{j \mathop = 0}^\infty z_{t - j} = z_t + z_{t - 1} + z_{t - 2} + \dotsb$

## Sources

$1$: Introduction:
$1.2$ Stochastic and Deterministic Dynamic Mathematical Models
$1.2.1$ Stationary and Nonstationary Stochastic Models for Forecasting and Control: Nonstationary models