Introduction

In the realm of time series analysis, understanding the concept of white noise is fundamental. White noise serves as a benchmark against which patterns and trends in time series data are assessed. This chapter delves into the definition and types of white noise, shedding light on independent white noise and normal (Gaussian) white noise. By comprehending the characteristics and implications of white noise, you will gain insights crucial for accurate time series modeling and analysis.

Defining White Noise

White noise is a fundamental concept in time series analysis, representing a series of uncorrelated random variables. In white noise, each data point is considered an independent and identically distributed (i.i.d.) random variable with a constant mean and variance. Essentially, white noise lacks any discernible pattern, trend, or structure, making it a crucial point of reference for identifying genuine signals in time series data.

Independent White Noise

Independent white noise refers to a sequence of random variables that are statistically independent of each other. This implies that the occurrence of one data point does not provide any information about the occurrence of another. In other words, the values in an independent white noise sequence do not exhibit any interdependence or correlation. This property makes independent white noise a foundational concept in statistical analysis, aiding in distinguishing between true signals and random fluctuations.

Normal (Gaussian) White Noise

Normal white noise, also known as Gaussian white noise, adheres to a Gaussian (normal) distribution. In this case, the random variables in the sequence follow a bell-shaped curve, with the majority of values concentrated around the mean, and fewer values deviating as you move away from the mean. The normal distribution is defined by its mean $(\mu)$ and standard deviation $(\sigma)$, which determine the shape and spread of the curve. Normal white noise is particularly important due to its prevalence in various natural and financial phenomena.

Example: Consider a daily stock price series that exhibits no clear upward or downward trend. Fluctuations in the stock price that seem random and unpredictable could be indicative of white noise. By identifying and separating white noise from true trends, analysts can make more accurate predictions and decisions.