subGaussian

SubGaussian refers to a class of random variables whose tail behavior is dominated by that of a Gaussian distribution. Specifically, a random variable X is called subGaussian if there exist positive constants such that the probability of X deviating from its mean by a certain amount declines at least as fast as the tail of a Gaussian distribution. This property implies strong concentration inequalities and is useful in probability theory, statistics, and machine learning.

Formally, a zero-mean random variable X is subGaussian if there exists a positive parameter σ such that for all t ≥ 0, the tail bound holds: P(|X| ≥ t) ≤ 2 exp(−t²/(2σ²)). Equivalently, X is subGaussian if its moment-generating function (MGF) satisfies E[exp(λX)] ≤ exp(λ²σ²/2) for all λ in ℝ. The parameter σ is called the subGaussian parameter or proxy variance.

SubGaussian variables include Gaussian random variables, bounded variables (such as those confined within a fixed interval), and many light-tailed distributions. They are extensively used in empirical process theory, concentration inequalities, and high-dimensional statistics to provide probabilistic bounds for sums of independent subGaussian variables.

The concept extends to vectors and processes, leading to the notions of subGaussian vectors and subGaussian processes, which generalize tail and concentration properties in higher dimensions or functional spaces. The utility of subGaussian variables lies in their ability to facilitate rigorous analysis of randomized algorithms, learning models, and statistical estimators, ensuring stability and robustness in various applications.