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

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

A random process \(\{ X_t : t \in T \}\) is called sub-gaussian if there is some constant \(c > 0\) such that

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 \(T \subseteq \R^n\),

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

where \(\Delta \seteq \max \{ d(t,t_0) : t \in T \}\).

A simple calculation gives the conclusion