Processindependence

Processindependence is a term used to describe a property of a collection of stochastic or computational processes in which the evolution of each process proceeds independently of the others. In probability theory, this is typically described as the independence of stochastic processes. A family {X^i_t : t ∈ T, i ∈ I} is independent if for every finite subset {i1, ..., ik} ⊂ I and every finite collection of time points t1, ..., tn ∈ T, the random vector (X^{i1}_{t1}, ..., X^{ik}_{tn}) has the product distribution of its marginal components. Equivalently, the sigma-algebras generated by the processes over the time domain are independent.

This property is important when modeling systems driven by multiple stochastic sources, such as network queues, reliability systems, or multi-factor financial models. For example, independent Poisson processes model separate arrival streams, and independent Brownian motions model independent sources of continuous noise. In computer science, processindependence can refer to the absence of interference between concurrent processes, beyond any explicitly shared inputs or synchronization constraints.

Independence has implications for analysis: it implies that expectations factor, variances add, and limit theorems simplify under independence of finite collections. However, independence does not follow from zero correlation; two processes can be uncorrelated yet dependent. It can also be weakened to conditional independence given a third process or sigma-algebra, or to independence on restricted time grids.

See also stochastic independence, mutual independence, sigma-algebra independence, and product probability spaces. Related concepts include the Markov property and martingales, which describe different structural aspects of stochastic processes.