I know that Seasonality and Trend violate the principle of stationarity, so before modelling the time series with many statistical models like AR, MA and ARMA it's important to remove those components to make time series stationary. When we remove Trend and Stationarity it just remains an Irregular component that we cannot predict because it's unpredictable by nature. My question is, how can we build a model with an irregular term ?
See this post.
The main idea behind ARIMA models is that after you remove the trend and the seasonality from the time series, what remains is not irregular.
What is left after making the time series stationary still has some additional structure to it besides the trend and the seasonality, and that structure can be modeled as an ARMA process.
This points to a major misconception about ARIMA models: Because they are an older model (first proposed in the 1970s) and are introduced at the beginning of various tutorials and chapters on time series, people assume that they are simple or basic models. ARIMA models are not. They are actually very complex. And your comment about
When we remove Trend and Stationarity it just remains an Irregular component that we cannot predict because it's unpredictable by nature. My question is, how can we build a model with an irregular term ?
is spot on. A lot of business time series are just "trend + seasonalities + noise" and trying to model them with ARIMA is not a good idea, but it is done very often nonetheless because of ARIMA's status in the literature, more so because of it being the right type of model to use.