Monotonicity and bounds of effective bandwidth ($\mathbb{E}X\le \log \mathbb{E}{e}^{sX}/s\le \mathrm{esssup}X$)

For $s>0$ in the finite moment-generating function domain and integrable $X$, Jensen inequality gives the lower bound and $X\le \mathrm{esssup}X$ almost surely gives the upper bound. The Holder inequality makes $K(s)=\log \mathbb{E}{e}^{sX}$ convex with $K(0)=0$. Thus $K(t)\le (t/s)K(s)$ for $0<t<s$, and the effective bandwidth is nondecreasing. It need not be strictly increasing: deterministic traffic gives a constant bandwidth. Infinite peaks or exponential moments require extended-value conventions.