Any examples of stochastic processes with continuous random variables and discrete time?

Question

What are the examples of stochastic processes with continuous random variables and discrete time? Also, is Brownian motion defined on $t = [0,\infty[$?

Answer 1

Three types of examples among many possibilities:

(1) Random walks (on the line, plane, etc.) with a uniformly distributed displacement left or right at each discrete step in time. The location at a given step is a continuous random variable.

(2) Many Gibbs samplers are discrete-time ergodic Markov chains with continuous values generated at each step. The chain is contrived so that the limiting distribution of the chain is the posterior distribution of a Bayesian model. By simulating the chain and looking at many simulated values near the end of a run, one can approximate the unknown posterior distribution. (One simple example here.)

(3) Metropolis-Hastings approximations usually involve random walks in multi-dimensional spaces. At each step a random displacement in the space is made and a candidate value (often continuous) is generated, the candidate value can be accepted or rejected according to some criterion. If rejected, the old value is replicated and a new displacement is tried. The result is a simulated sample from a multivariate distribution. (Wikipedia discussion.)