Definition:Probability Limit/Lower

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Let $T$ be a time series.

Let $S$ denote the range of $T$.

Let $L$ denote the set of lead times of $T$.

Let $\hat z_t$ be a forecast function on $L$.

Let $\map {\hat z_t} l$ denote the forecast value of the observation at the timestamp of lead time $l$.

Let $\Delta_p$ be a probability limit for $\map {\hat z_t} l$.

The lower probability limit of $\map {\hat z_t} l$ is the value $\map {\hat z_t} l - \Delta_p$.


$1$: Introduction:
$1.1$ Four Important Practical Problems:
$1.1.1$ Forecasting Time Series