Matthias Löwe

Analysis of the expected shortfall of aggregate dependent risks

We consider d identically and continuously distributed dependent risks X1 ,… , X d . Our main result is a theorem on the asymptotic behaviour of expected shortfall for the aggregate risks: there is a constant cd such that for large u we have d ii 11 == EX X u uc ii #+ dd !! 8 B . Moreover we study diversification effects in two dimensions, similar to our Value-at-Risk studies in [2].

A large deviation principle form-variate von Mises-statistics and U-statistics

A large deviation principle form-variate von Mises-statistics and U-statistics with a kernel function satisfying natural moment conditions is proved. Sanovs large deviation result for the empirical distribution function and two fundamental conservation principles in large deviation theory are the main tools. The rate functions are “drawback”-entropy functionals.

Reconstruction of sceneries with correlated colors

Matzinger (Random Structure Algorithm 15 (1999a) 196) showed how to reconstruct almost every three color scenery, that is a coloring of the integers with three colors, by observing it along the path of a simple random walk, if this scenery is the outcome of an i.i.d. process. This reconstruction needed among others the transience of the representation of the scenery as a random walk on the three-regular tree T3. Den Hollander (private communication) asked which conditions are necessary to ensure this transience of the representation of the scenery as a random walk on T3 and whether this already suffices to make the reconstruction techniques in Matzinger (1999a) work. In this note we answer the latter question in the affirmative. Also we exhibit a large class of examples where the above-mentioned transience holds true. Some counterexamples show that in some sense the given class of examples is the largest natural class with the property that the representation of the scenery as a random walk is transient.

Simulated annealing with time-dependent energy function via Sobolev inequalities

We analyze the simulated annealing algorithm with an energy function Ut that depends on time. Assuming some regularity conditions on Ut (especially that Ut does not change too quickly in time), and choosing a logarithmic cooling schedule for the algorithm, we derive bounds on the Radon-Nikodym density of the distribution of the annealing algorithm at time t with respect to the invariant measure [pi]t at time t. Moreover, we estimate the entrance time of the algorithm into typical subsets V of the state space in terms of [pi]t(Vc).

Mixing times for the Swapping Algorithm on the Blume-Emery-Griffiths model

We analyze the so called Swapping Algorithm, a parallel version of the well-known Metropolis-Hastings algorithm, on the mean-field version of the Blume-Emery-Griffiths model in statistical mechanics. This model has two parameters and depending on their choice, the model exhibits either a first, or a second order phase transition. In agreement with a conjecture by Bhatnagar and Randall we find that the Swapping Algorithm mixes rapidly in presence of a second order phase transition, while becoming slow when the phase transition is first order.

On the storage capacity of the Hopfield model with biased patterns

We introduce a form of the Hopfield model that is able to store an increasing number of biased i.i.d. patterns (it is well known that the standard Hopfield model fails to work properly in this context). We show that this new form of the Hopfield model with N neurons can store (N)/(/spl gamma/ log N) or /spl alpha/N biased patterns (depending on which notion of storage is used). The quantity /spl gamma/ increases with an increasing bias of the patterns, while /spl alpha/ decreases when the bias gets large.

On a Model of Associative Memory with Huge Storage Capacity

In Krotov et al. (in: Lee (eds) Advances in Neural Information Processing Systems, Curran Associates, Inc., Red Hook, 2016) Krotov and Hopfield suggest a generalized version of the well-known Hopfield model of associative memory. In their version they consider a polynomial interaction function and claim that this increases the storage capacity of the model. We prove this claim and take the ”limit” as the degree of the polynomial becomes infinite, i.e. an exponential interaction function. With this interaction we prove that model has an exponential storage capacity in the number of neurons, yet the basins of attraction are almost as large as in the standard Hopfield model.

The Effect of System Load on the Existence of Bit Errors in CDMA With and Without Parallel Interference Cancelation

In this correspondence, we study a lightly loaded code-division multiple-access (CDMA) system with and without multistage hard- and soft-decision parallel interference cancelation (HD-PIC and SD-PIC). Throughout this paper we will only consider the situation of a noiseless channel, equal powers and random spreading codes. For the system with no or a fixed number of steps of interference cancelation, we give a lower bound on the maximum number of users such that the probability for the system to have no bit-errors converges to one. Moreover, we investigate when the matched filter system, where parallel interference cancelation is absent, has bit errors with probability converging to one. This implies that the use of HD-PIC and SD-PIC significantly enhances the number of users the system can serve

A Comparative Study of Sparse Associative Memories

We study various models of associative memories with sparse information, i.e. a pattern to be stored is a random string of 0s and 1s with about