sub-Gaussian process

Definition

A collection of zero-mean random variables $\{X_{\theta }:\theta \in \Theta \}$ is a sub-Gaussian process wrt to a metric $\rho$ if

$$\mathbb{E}[\exp (\lambda (X_{\theta }-X_{\varphi }))]\le \exp ({\lambda }^2{\rho }^2(\theta ,\varphi )/2),$$

for all $\theta ,\varphi \in \Theta$.

The Rademacher process and the canonical Gaussian process (see Rademacher complexity and Gaussian complexity) are examples of sub-Gaussian processes.

As in the case of tail bounds for sub-Gaussian random variables (bounded scalar concentration), using the Chernoff method we see that the above definition is equivalent to

$$\mathbb{P}(|X_{\theta }-X_{\varphi }|\ge ϵ)\le 2\exp \left(-\frac{t^2}{2{\rho }^2(\theta ,\varphi )}\right).$$