Continuous-time Mallows processes

Source abstract

Abstract In this article, we introduce Mallows processes , defined to be continuous-time cadlag processes with Mallows distributed marginals. We show that such processes exist and that they can be restricted to have certain natural properties. In particular, we prove that there exist regular Mallows processes, defined to have their inversion numbers normal upper I normal n normal v Subscript j Baseline left parenthesis sigma right parenthesis equals StartAbsoluteValue StartSet i element of left bracket j minus 1 right bracket colon sigma left parenthesis i right parenthesis greater than sigma left parenthesis j right parenthesis EndSet EndAbsoluteValue I n v j ( σ ) = | { i ∈ [ j − 1 ] : σ ( i ) > σ ( j ) } | ${\mathrm{Inv}}_j(\sigma )=|\{i\in [j-1] \, : \, \sigma (i)\gt \sigma (\,j)\}|$ be independent increasing stochastic processes with jumps of size 1 1 $1$ . We further show that there exists a unique Markov process which is a regular Mallows process. Finally, we study properties of regular Mallows processes and show various results on the structure of these objects. Among others, we prove that the graph structure related to regular Mallows processes looks like an expanded hypercube where we stacked k k $k$ hypercubes on the dimension k element of left bracket n right bracket k ∈ [ n ] $k\in [n]$ ; we also prove that the first jumping times of regular Mallows processes converge to a Poisson point process; we eventually provide results regarding the Markov property of the discrete jumping chain corresponding to a Mallows process.