The invariance principle for Banach space valued random variables

We extend the invariance principle to triangular arrays of Banach space valued random variables, and as an application derive the invariance principle for lattices of random variables. We also point out how the q-dimensional time parameter Yeh-Wiener process is naturally related to a one dimensional time Wiener process with an infinite dimensional Banach space as a state space.     

On the Mean Pair Correlation Function of ±J Ising Spin Glasses

We provide a formula and an upper bound for the average over the disorder of the pair correlation function of ±J Ising spin glasses by using the symmetries of the system, We show the decay of the mean spin pair correlation function when the proportion of antiferromagnetic bonds is larger than the critical parameter associated with the pair dissociation phase transition.     

The meaning of general covariance of einstein’s theory of gravity

The fundamental symmetry of Einstein’s theory of gravity is Lorentz-invariance which leads to a well defined energy-momentum tensor. This is also true for Maxwell’s theory of electromagnetism which has an additional symmetry due to its spin one, restmass zero character. Similarly, the spin two, restmass zero character of the gravitational field leads to an additional gauge symmetry that happens to be isomorphic to the concept of general covariance. The gauge-covariant energy-momentum tensor for gravitational interactions vanishes identically.     

Martingales and the Robbins-Monro procedure in D[0, 1]

The Robbins-Monro procedure for recursive estimation of a zero point of a regression function f is investigated for the case f defined on and with values in the space D[0, 1] of real-valued functions on [0, 1] that are right-continuous and have left-hand limits, endowed with Skorohod's J₁-topology. There are proved an a.s. convergence result and an invariance principle where the limit process is a Gaussian Markov process with paths in the space of continuous C[0, 1]-valued functions on [0, 1]. At first the case f(x) ≡ x, i.e., the case of a martingale in D[0, 1], is treated and by this then the general case. An application to an initial value problem with only empirically available function values is sketched.     

Premium principles and translation invariance

Premium principles τ are examined with respect to the property of translation invariance, that means τ(X + c) = τ(X) + c holds for all real c (and all risks X). It is shown that this condition of translation invariance is highly overdetermined: For all known principles the range of c can be reduced to a large extent from all the reals to only two (or one) single values of c respectively.     

A note on a theorem of Berkes and Philipp for dependent sequences

We improve an almost sure invariance principle for f-mixing sequences of real random variables with finite (2 + δ)th moment (0 < δ 2) due to Berkes and Philipp (1979). The exponent of the slow logarithmic rate of mixing in that theorem is relaxed from 160/δ to (1 + ε)(1 + 2/δ) and the undetermined quantities a_N are replaced by Nσ² for some σ > 0.     

A note on solving nonlinear equations and the natural criterion function

We consider the natural criterion function for controlling a damped Newton iteration used for solving systems of nonlinear equations. An example is given which shows that the method can cycle in an undesirable manner, not possible when using the more conventional least-squares criterion function. Despite this, the natural criterion function is of value in particular applications. This point is illustrated using a system derived from a nonlinear two-point boundary-value problem.     

The central limit theorem for time series regression

The central limit problem is considered for a simple regression, where the residuals, x(n), are stationary and the sequence regressed on y^(N)(n), may depend on the number of observations, N, to hand. Two situations are considered, one where the residual is generated by a linear process (i.e. the best linear predictor is the best predictor) and the more general situation where that is not so. Two types of conditions are needed, the first of which limits the contribution of any individual y^(N)(n) and the second of which relates to the mixing properties of x(n). If ε(n) is the linear innovation sequence, in the linear case, with $\text{lim}{}_{\mathrm{k}}\text{→∞}\text{E}\text{(ε(n)}{}^2\text{∥}\text{F}{}_{\mathrm{n?k}}\text{)=}\text{F}{}^2\text{,}\text{F}{}_{\mathrm{n}}$ being the associated family of o-algebra, then the central limit theorem holds under minimal conditions on y^(N)(n). Under sligthly stronger conditions on y^(N)(n) and for x(n) weakly mixing this theorem and associated theorems, are shown to hold under further fairly weak conditions on the dependence of x(n) on its past.