Markov chains generated by maximizing components of multidimensional extremal processes
Let \(X_ i=(X_{i1},X_{i2},...,X_{im})\), \(i=1,2,..\). be a sequence of independent identically distributed random vectors. Define the stochastic process \(U_{nj}(t)=\max_{i\leq [nt]}X_{ij}\). Suppose that \(U_ n(t)=(U_{n1}(t),...,U_{nm}(t))\), suitably normed converges to a process \(\{\) Z(t)\(\}\). In this case \(\{\) Z(t)\(\}\) belongs to the class of multidimensional inhomogeneous extremal processes. Define the finite state process \(\{\) J(t)\(\}\) with \(J(t)=j\) if \(Z_ j(t)=\max_{k}Z_ k(t).\) In the case when \(\{Z_ k(t)\}\), \(k=1,...,m\) are independent extremal processes \(\{\) J(t)\(\}\) is an inhomogeneous Markov chain. By means of the marginal distribution \(F_ t(x)\) formulae for the transition intensities, the transition and state probabilities are obtained and the relation between them is determined. On condition \(J(s)=i\) the distribution of the excursion time, i.e. during which \[ \inf_{s\leq \tau \leq t}(Z_ i(\tau)-\max_{k\neq i}Z(\tau))>0 \] is obtained. The results derived are of particular interest in behavioral sciences, since they provide a framework for analyzing individual discrete decisions over time.
scientific article; zbMATH DE number 4046812