Kdependent

Kdependent, commonly written as k-dependent, is a concept in probability theory describing a localized dependence structure among a family of random variables. A collection {X_i} indexed by an ordered index set is k-dependent if any two finite subcollections that are separated by at least k indices are independent. In practical terms, variables that are sufficiently far apart by more than k do not influence each other, while variables within a distance of k may be dependent.

Equivalent formulations: If A and B are finite subsets of the index set with max(A) < min(B) − k, then the sets {X_i: i in A} and {X_j: j in B} are independent. A stationary k-dependent process is one whose distribution is invariant under shifts and satisfies this separation property for all indices.

Notes: There are variant conventions depending on indexing and whether the parameter is described as a separation in time or in index distance. In many texts, k-dependent is paired with m-dependent terminology, using the same idea with different letter choices.

Examples: A process where blocks of length k are generated independently, with no cross-block dependence, is k-dependent. An independent and identically distributed (i.i.d.) sequence is k-dependent for every k, since all finite subcollections are independent regardless of separation.

Applications: K-dependent models appear in the probabilistic method, random graphs, and the analysis of local algorithms and percolation, where long-range independence is not assumed but independence across sufficiently separated blocks is useful for analysis.

See also: m-dependence, dependency graphs, mixing conditions, stationary processes.