Centered 1-sub-Gaussian reward

Let $X$ be an integrable real random variable on a probability space, and write $\mu =\mathbb{E}[X]$. The reward $X$ is 1-sub-Gaussian if for every $\lambda \in \mathbb{R}$,

$$\mathbb{E}\text{\hspace{0.17em}⁣}[\exp \text{\hspace{0.17em}⁣}(\lambda (X-\mu ))]\le \exp \text{\hspace{0.17em}⁣}\left(\frac{{\lambda }^2}{2}\right)\mathrm{.}$$

The expectation in this condition is the nonnegative Lebesgue integral of the displayed exponential; the inequality requires it to be finite for every real $\lambda$.

Random elements and real random variables: a real random variable is a measurable map into $(\mathbb{R},\mathcal{B}(\mathbb{R}))$.

Expectation of a nonnegative or integrable random variable: expectation is the Lebesgue integral against the underlying probability measure, finite for an integrable real random variable.

The real exponential function and the number $e$ by a power series: $\exp (x)$ is the real exponential function defined for every $x\in \mathbb{R}$.

Tor Lattimore and Csaba Szepesvári, Bandit Algorithms: Definition 5.2, PDF p. 86 / printed p. 77; centered-noise convention, PDF p. 88 / printed p. 79.

Sébastien Bubeck and Nicolò Cesa-Bianchi, Regret Analysis of Stochastic and Nonstochastic Multi-armed Bandit Problems: Equation (2.2), Chapter 2 §2.2, PDF p. 13 / printed p. 10.