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;
GMP-->MP;
OUP[Ornstein-Uhlenbeck Process]-->GMP;
PPP[Poisson Point Process]-->PP;
PPP-->MP;
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.
https://www.whoi.edu/cms/files/lecture06_21268.pdf
Point processes
Definition. Point process.
Definition. Poisson point process.
Definition. Cox process.
Definition. Hawkes process.
https://arxiv.org/abs/1507.02822
Gaussian processes
Definition. Gaussian process.
Kernel.
https://en.wikipedia.org/wiki/Covariance_function
https://www.cs.toronto.edu/~duvenaud/cookbook/
$$ \Prc{\vec f_*}{\vec x_*, \vec x, \vec y} $$
To do. Kernel trick.
http://www.gaussianprocess.org/gpml/chapters/RW4.pdf
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)[https://en.wikipedia.org/wiki/Gaussian_process_regression] is also known as Kriging and Wiener–Kolmogorov prediction.