Distances on the tropical line determined by two points

Let $p',q'\in R^n$. Write $p'\sim q'$ if $p'-q'$ is a multiple of $(1,\ldots,1)$. Two different points $p$ and $q$ in $R^n/\sim$ uniquely determine a tropical line $L(p,q)$, passing through them, and stable under small perturbations. This line is a balanced unrooted semi--labeled tree on $n$ leaves. It is also a metric graph. If some representatives $p'$ and $q'$ of $p$ and $q$ are the first and second columns of some real normal idempotent order $n$ matrix $A$, we prove that the tree $L(p,q)$ is described by a matrix $F$, easily obtained from $A$. We also prove that $L(p,q)$ is caterpillar. We prove that every vertex in $L(p,q)$ belongs to the tropical linear segment joining $p$ and $q$. A vertex, denoted $pq$, closest (w.r.t tropical distance) to $p$ exists in $L(p,q)$. Same for $q$. The distances between pairs of adjacent vertices in $L(p,q)$ and the distances \dd(p,pq), \dd(qp,q) and \dd(p,q) are certain entries of the matrix $|F|$. In addition, if $p$ and $q$ are generic, then the tree $L(p,q)$ is trivalent. The entries of $F$ are differences (i.e., sum of principal diagonal minus sum of secondary diagonal) of order 2 minors of the first two columns of $A$.Comment: New corrected version. 31 pages and 9 figures. The main result is theorem 13. This is a generalization of theorem 7 to arbitrary n. Theorem 7 was obtained with A. Jim\'enez; see Arxiv 1205.416

Modelling long-run trends and cycles in financial time series data

Copyright @ 2012 Wiley Publishing Ltd. This is the accepted version of the following article: "Modelling long-run trends and cycles in financial time series data", Journal of Time Series Analysis, 34(3), 405-421, 2013, which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1111/jtsa.12010/abstract.This article proposes a general time series framework to capture the long-run behaviour of financial series. The suggested approach includes linear and segmented time trends, and stationary and non-stationary processes based on integer and/or fractional degrees of differentiation. Moreover, the spectrum is allowed to contain more than a single pole or singularity, occurring at both zero but non-zero (cyclical) frequencies. This framework is used to analyse five annual time series with a long span, namely dividends, earnings, interest rates, stock prices and long-term government bond yields. The results based on several likelihood criteria indicate that the five series exhibit fractional integration with one or two poles in the spectrum, and are quite stable over the sample period examined.Ministerio de Ciencia y Tecnologia and Jeronimo de Ayanz project of the Government of Navarra