# Examples, applications, Gaussian processes (Mon, Jan 09)

• 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

$$$\label{eq:sg-tail} \P[X_s - X_t > \lambda] \leq \exp\left(- c\lambda^2/d(s,t)^2\right)$$$

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

$\E \max_{t \in T} X_t \lesssim \Delta \sqrt{\log |T|}\,.$