A random variable $X$ is sub-Gaussian if there exists a $b>0$ such that for all $t \in \mathbb{R}$ we have $$\mathbb{E}(\exp(tX)) \leq \exp(b^2t^2/2).$$
According to some sources online such as here we have that $\mathbb{E}(X)=0$ (see proposition 2.1). Another source (see definition 1.2) defines random variables as sub Gaussian if it satisfies the above equation and that $\mathbb{E}(X)=0$.
I've been reading into a variant of the multi armed bandit problem recently called rotting bandits. The latest version of the preprint is here, where the authors state:
When arm $i$ is pulled for the $n^{th}$ time, the agent receives a time-independent, $σ^2$ sub-Gaussian random reward, $r_t$, with mean $\mu_i(n).$
Another paper on the more general problem of Multi-Armed Bandits with Non-Stationary Rewards references the previously linked paper saying the authors analyze
the setting where the rewards are $\sigma^2$ sub-Gaussian rewards with mean $\mu_i^c+t^{-\theta_i^*}$ at time $t$ for $\theta_i^* \in \Theta=\{\theta_1\,\theta_2,\ldots\}.$
My confusion here is that the reward in the rotting bandit scenario are sub-Gaussian with potentially non-zero means, contradiction my linked reading into sub-Gaussian distributions. Can anyone identify where I have made a mistake here?
Note: I'm aware Arxiv is not a peer-reviewed platform however the authors come from reputable institutions.
Best Answer
It is standard in the bandit literature to abuse notation by considering a random variable $X$ to be $\sigma$-subgaussian if the noise $X - \mathbb{E}[X]$ is $\sigma$-subgaussian.
See the note on page 78 of Tor Lattimore and Csaba Szepesvari's book