A random process is a collection \(\{X_t : t \in T\}\) of random variables. For the next few lectures, we will considered symmetric processes, i.e., those where \(X_t\) and \(-X_t\) have the same law for every \(t \in T\).
We equip the index set \(T\) of such a process with the natural \(\ell_2\) metric
\[d(s,t) \seteq \sqrt{\E(X_s-X_t)^2}.\]A fundamental example is that of a Gaussian process, where the family \(\{X_t : t \in T\}\) is a jointly normal family of random variables. When \(T\) is finite, any such family can be described as follows: \(T \subseteq \R^n\) for some \(n \geq 1\), and \(X_t = \langle g,t\rangle\), where \(g=(g_1,\ldots,g_n)\) is a standard \(n\)-dimensional gaussian, i.e., the coordinates \(g_1,\ldots,g_n\) are i.i.d. \(N(0,1)\) random variables.
In this case the geometry of the metric space \((T,d)\) is Eucliean, since \(d(s,t) = \|s-t\|_2\).
Also note that the metric \(d(s,t)\) completely describes the covariance structure of the process, and therefore characterizes it (this is a special property of jointly normal families of random variables).
We will be concerned with extrema of such a process, and to that end we study the quantity
\[\E \max_{t \in T} X_t\,.\]A random process \(\{ X_t : t \in T \}\) is called sub-gaussian if there is some constant \(c > 0\) such that
\[\begin{equation}\label{eq:sg-tail} \P[X_s - X_t > \lambda] \leq \exp\left(- c\lambda^2/d(s,t)^2\right) \end{equation}\]In particular, Gaussian processes are sub-gaussian with \(c=1/2\).
Another example is a Bernoulli process, i.e. a family of random variables defined thusly: For some subset \(T \subseteq \R^n\),
\[\left\{ \e_1 t_1 + \e_2 t_2 + \cdots + \e_n t_n : t = (t_1,\ldots,t_n) \in T \right\},\]where \(\e_1,\ldots,\e_n \in \{-1,1\}\) are i.i.d. uniformly random signs.
Diameter bound: When the index set \(T\) is finite, we can use the tail bound \eqref{eq:sg-tail} to give a simple upper bound on \(\E \max_{t \in T} X_t\).
Fix some \(t_0 \in T\) and note that \(\E \max_{t \in T} X_t = \E \max_{t \in T} (X_t-X_{t_0})\). Now applying a union bound gives
\[\P\left[\max_{t \in T} (X_t-X_{t_0}) > \lambda\right] \leq |T| \exp\left(- c\lambda^2/\Delta^2\right)\,,\]where \(\Delta \seteq \max \{ d(t,t_0) : t \in T \}\).
A simple calculation gives the conclusion
\[\E \max_{t \in T} X_t \lesssim \Delta \sqrt{\log |T|}\,.\]