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The projective line with respect to a local field is the boundary of the Bruhat-Tits tree associated to the field, much in the same way as the real projective line is the boundary of the upper half-plane. In both cases we may consider the horocycles with respect to the point at infinity. These horocycles are exactly the horizontal lines {y = a} with a > 0 in the real case, while in the case of a local field the horocycles may be thought of as discretizations of the field obtained by collapsing to a point each ball of a given radius.In this paper we exploit this geometric parallelism to construct symmetric α-stable random variables on the real line and on a local field by completely analogous procedures. In the case of a local field the main ingredient is a drifted random walk on the tree. In the real case the random walk is replaced by a drifted Brownian motion on the hyperbolic halfplane. In both cases the random processes are invariant under the automorphisms of the tree and the hyperbolic half-plane, respectively, that fix the point at infinity.These random processes determine hitting distributions on the horocycles which, in a sense to be specified, are shown to be in the domain of attraction of α-stable symmetric random variables. In both cases the exponent of α-stability is related by an explicit formula to the drift coefficient.

We describe a relationship between globalizations of local holomorphic actions on Stein manifolds induced by global actions of certain non-compact Lie groups, and holomorphic fiber bundles with smooth Stein base and fiber and connected structure group. To this end we prove a univalence result for particular Stein Riemann domains with a free and properly discontinuous action of a discrete group of biholomorphisms. We then derive some consequences on the existence of Stein globalizations. Subject Classification (1991): 32M05, 32L05, 32D10, 32E10. Mathematics

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This paper analyzes some aggregation aspects of the procedure for constructing a composite index on a multidimensional socio-economic phenomenon such as development, the main focus being on the unbalance among individual dimensions. First a theoretical framework is set up for the unbalance adjustment of the index. Then an aggregation function is proposed that takes unbalance among development dimensions into account; a separate index is also introduced that measures the unbalance itself. Finally the dataset of the Index of African Governance for the year 2007 is used to test this method and compare it against the weighted arithmetic mean of variables with relation to the measured values of unbalance, yielding significantly different results for ratings and rankings, which in addition show negative correlations with the unbalance adjustment values. The changes ensuing from the adjustment are commented for some countries. © 2012 Springer Science+Business Media B.V