Stochastic processes

graph BT;
    SP[Stochastic Process];
    GSP[Gaussian Stochastic Process]-->SP;
    PP[Point Process]-->SP;
    RF[Random Field]-->SP;
    MRF[Markov Random Field]-->RF;
    IM[Ising Model]-->MRF;
    CTP[Continuous Time Process]-->RF;
    DTP[Discrete Time Process]-->SP;
    MP[Markov Process]-->CTP;
    MP[Markov Process]-->MRF;
    MC[Markov Chain]-->DTP;
    GMP[Gauss-Markov Process]-->GSP;
    OUP[Ornstein-Uhlenbeck Process]-->GMP;
    PPP[Poisson Point Process]-->PP;
    LP[Lévy process]-->MP;
    WP[Wiener Process]-->LP;
    HP[Hawkes Process]-->PP;

For the length of this chapter, assume we are give a probability space $(Ω, Σ, \operatorname{Pr})$.

Definition. Given a probability space $(Ω, Σ, \operatorname{Pr})$, a measurable space $(Ω_X, Σ_X)$ and a set $T$, a stochastic process is a function $X: T \times Ω → Ω_X$ such that for each $t ∈ T$ the restricted function $X\p{t, \dummyarg}: Ω → Ω_X$ is a random variable.

Note. The set $T$ is called the index set or parameter set of the stochastic process. The set $Ω_X$ is called the state space. Given $ω ∈ Ω$ the restricted function $X\p{\dummyarg, ω}: T → Ω_X$ is called a sample function, realization, sample path, trajectory, path function, or path.

Depending on the nature of the index set $T$, a stochastic process can be called many things:

Definition. If the index set $T$ is a topological space then $X$ is a random field.

Note. Nearly all stochastic processes are random fields by definition, but the term is generally reserved for when $T$ has more than one dimension.

Definition. If the index set $T$ is a countable ordered set then $X$ is a discrete-time process.

Definition. If the index set $T$ is an interval of $\R$ then $X$ is a continuous-time process.

Definition. If the index set $T$ are the vertices of a graph and satisfies To do. properties, then $X$ is a Markov random field.

Note. Similarly to the above definitions, a stochastic process can also be named based on the state space $Ω_X$. discrete stochastic process, integer-valued stochastic process, real-valued stochastic process, $n$-dimensional vector process.

Definition. Stationary process.

Definition. Markov process.

Definition. Martingale.

Definition. Lévy process.

Definition. Bernoulli process.

Definition. Random walk.

Definition. Wiener process.

To do. Theorem. Every continuous-time independent-increment process is a Gaussian process. Proof using central limit theorem.

Point processes

Definition. Point process.

Definition. Poisson point process.

Definition. Cox process.

Definition. Hawkes process.

Gaussian processes

Definition. Gaussian process.


$$ \Prc{\vec f_*}{\vec x_*, \vec x, \vec y} $$

To do. Kernel trick.

Definition. Gauss-Markov process.

Definition. Matérn kernel.

Definition. Squared exponential kernel aka RBF kernel. Special case of Matérn $\nu = \infty$.

Definition. Rational quadratic kernel.

Definition. Ornstein–Uhlenbeck process. Aka absolute exponential. Special case of Matérn with $\nu = \frac 12$.

Gaussian Process Regression

Note. (Gaussian process regression)[] is also known as Kriging and Wiener–Kolmogorov prediction.

Remco Bloemen
Math & Engineering