Concentration inequalities for stochastic processes

Question

I understand this is a rather vague question, but I am curious if there exist any generalization of, e.g. Hoeffding's inequality or Chernoff's inequality to the case of stochastic processes? I am in particular interested in upper bounding the following term $$P(\sup_{t\in T}\vert X(t)-EX(t) \vert \ge c)$$ where $t$ is a continuous (or discrete, if necessary) parameter where $T \subset \mathbb{R}$, $EX(t)$ is the expectation of the random variable $X(t)$, and $X$ and $T$ can be arbitrarily nice, for example, perhaps for Gaussian processes and $T$ compact we can say something about the above term?

Answer 1

Let us also assume $X$ is separable (e.g. continuous). Then $\sup_{t\in T} |X_t| = \sup_{n\ge 1} |X_{t_n}|$, in other words, it is enough to look at sequences.

A useful fact:

Theorem (Fernique-Landau-Marcus-Shepp) Let $(Y_n,n\ge1)$ be a Gaussian sequence, and $Y^* = \sup_{n\ge 1} |Y_n|$. If $\mathbb P(Y^*<\infty)>0$, then $\mathbb P(Y^*<\infty)=1$, moreover, for any $c<(2\sigma^2)^{-1}$, where $\sigma^2:=\sup_{n\ge 1} \operatorname{var}(Y_n)$, $$ \mathbb{E}[e^{c(Y^*)^2}]<\infty. \tag{1} $$

It follows from $(1)$ that $$ \limsup_{x\to+\infty}\frac{1}{x^2} \log \mathbb P(Y^* \ge x) \le -\frac{1}{2\sigma^2}. $$

Since, clearly, for any $x>0$, $$ \mathbb P(Y^* \ge x)\ge \mathbb P(\sigma|Z|\ge x), $$ where $Z$ is standard Gaussian, it follows that $$ \lim_{t\to\infty}\frac{1}{x^2} \log \mathbb P(Y^* \ge x) = -\frac{1}{2\sigma^2}. $$

Going back to $X$, in view of separability, $$ \lim_{x\to\infty}\frac{1}{x^2} \log \mathbb P\left(\sup_{t\in T} |X_t| \ge x\right) = -\frac{1}{2\sigma^2}\tag{2} $$ with $\sigma^2 = \sup_{t\in T} \operatorname{var}(X_t)$ provided that $X$ is separable and bounded. (Note that this is stated in Marcus-Shepp , but it seems that the authors forgot to explain this.)

The statement $(2)$ is "large-deviation" type and asymptotic, i.e. you don't have a precise bound for the probability. In order to get one, you need to impose some assumptions on the mean (which can be assumed zero for your question) and covariance of $X$. There are many works by Marcus, Pisier, Fernique, Talagrand, and others; you can start by looking in Google Scholar which articles cite Marcus-Shepp.