A note on statistical models for individual hedge fund returns

In recent years, a large number of research papers and monographs on the analysis of hedge fund returns have been published. Typically, the authors of these studies implicitly or explicitly treat monthly returns of hedge funds as independent and identically distributed observations. The Hedge Fund Index might be able to serve that role. But the returns of an individual hedge fund are not like that. They behave autoregressively depending on the time periods. This stochastic behavior should be modeled as a combined/regime switching stochastic process of two processes: i.i.d. process and autoregressive process. This paper first depicts the autoregressiveness of hedge fund returns. Then we introduce our statistical model for returns of an individual hedge fund and then, with our retrospective view, we perform several data analyses for individual hedge funds’ return data.     

A note on circular dimension

In [1] Feinberg conjectures that the maximum circular dimension of all graphs having n vertices is attained by a complete partite graph. In this note we show that this is not so.     

A note on reduction of the number of parameters in linear statistical models

In certain settings the mean response is modeled by a linear model using a large number of parameters. Sometimes it is desirable to reduce the number of parameters prior to conducting the experiment and prior to the actual statistical analysis. Essentially, it means to formulate a simpler approximate model to the original “ideal” one. The goal is to find conditions (on the model matrix and covariance matrix) under which the reduction does not influence essentially the data fit. Here we try to develop such conditions in regular linear model without and with linear restraints. We emphasize that these conditions are independent of observed data.     

A note on the dimension of a poset

It is shown that the dimension of a poset is the smallest cardinal number such that there is an embedding of the poset into a strict product of linear orders.     

A note on multifractal packing dimension of measures

The relations between the multifractal packing dimension of Borel probability measures and the asymptotic behavior of the function φ*(x) = lim sup r→0 logν(B(x,r)) - qlogμ(B(x,r)) logr are discussed and some applications are given.     

A note on APN permutations in even dimension

APN permutations in even dimension are vectorial Boolean functions that play a special role in the design of block ciphers. We study their properties, providing some general results and some applications to the low-dimension cases. In particular, we prove that none of their components can be quadratic. For an APN vectorial Boolean function (in even dimension) with all cubic components we prove the existence of a component having a large number of balanced derivatives. Using these restrictions, we obtain the first theoretical proof of the non-existence of APN permutations in dimension 4. Moreover, we derive some constraints on APN permutations in dimension 6.     

A note on attractors with finite fractal dimension

Conditions for the finite fractal dimension of precompact invariantsets are formulated and exponential attractors for discreteand continuous dynamical systems are constructed.     

A note on equi-integrability in dimension reduction problems

In the framework of the asymptotic analysis of thin structures, we prove that, up to an extraction, it is possible to decompose a sequence of ‘scaled gradients’ (where is the gradient in the k-dimensional ‘thin variable’ x _β) bounded in (1 < p < + ∞) as a sum of a sequence whose p-th power is equi-integrable on Ω and a ‘rest’ that converges to zero in measure. In particular, for k = 1 we recover a well-known result for thin films by Bocea and Fonseca (ESAIM: COCV 7:443–470; 2002).      

A note on GK dimension of skew polynomial extensions

Let be a finitely generated commutative domain over an algebraically closed field , an algebra endomorphism of , and a -derivation of . Then if and only if is locally algebraic in the sense that every finite dimensional subspace of is contained in a finite dimensional -stable subspace.

Similarly, if is a finitely generated field over , a -endomorphism of , and a -derivation of , then if and only if is an automorphism of finite order.

     

A note on the Brouwer dimension of chainable spaces

A general method produces from a compact Hausdorff space S a compact Hausdorff space T with IndT=IndS+1. We show that if S is chainable, then T is also chainable while DgT<IndT, where Dg denotes dimensionsgrad, the dimension in the original sense of Brouwer. This leads to a chainable, first countable, separable space Xn with DgXn<IndXn=n for each integer n>1.     

A note on the dimension of an orbital subspace

This note contains a dimension formula for an orbital subspace in a symmetry class of tensors corresponding to an irreducible character λ of a subgroup G of S_m. An algorithm for choosing a basis is also described.