A random process is a collection {Xt:t∈T} of random variables. For the next few lectures, we will considered symmetric processes, i.e., those where Xt and −Xt have the same law for every t∈T.
We equip the index set T of such a process with the natural ℓ2 metric
d(s,t) := √E(Xs−Xt)2.A fundamental example is that of a Gaussian process, where the family {Xt:t∈T} is a jointly normal family of random variables. When T is finite, any such family can be described as follows: T⊆Rn for some n≥1, and Xt=⟨g,t⟩, where g=(g1,…,gn) is a standard n-dimensional gaussian, i.e., the coordinates g1,…,gn 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)=‖.
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|}\,.