Random Walks in Clifford Algebras of Arbitrary Signature as Walks on Directed Hypercubes

#### R. Schott, G.S. Staples

2008, v.14, Issue 4, 515-542

ABSTRACT

Given a Clifford algebra of arbitrary signature $\mathcal{C}\ell_{p,q}$, $p+q=n$, multiplicative random walks are induced by sequences of independent, uniformly distributed random variables taking values in the unit basis vectors and paravectors in the algebra. These walks can be viewed as random walks on "directed hypercubes". Properties of such multiplicative walks are investigated. Sequences of multiplicative walks are then used to induce additive walks on the algebra. Limit theorems for these walks are then developed.

Keywords: Clifford algebras,random walks,Markov chains,hypercubes