Totally Ergodic Generalised Matrix Equilibrium States have the Bernoulli Property

We show that every totally ergodic generalised matrix equilibrium state is $$\psi $$ ψ -mixing with respect to the natural partition into cylinders and hence is measurably isomorphic to a Bernoulli shift in its natural extension. This implies that the natural extensions of ergodic generalised matrix equilibrium states are measurably isomorphic to Bernoulli processes extended by finite rotations. This resolves a question of Gatzouras and Peres in the special case of self-affine repelling sets with generic translations.

How linear reinforcement affects Donsker’s theorem for empirical processes

A reinforcement algorithm introduced by Simon (Biometrika 42(3/4):425–440, 1955) produces a sequence of uniform random variables with long range memory as follows. At each step, with a fixed probability $$p\in (0,1)$$ p ∈ ( 0 , 1 ) , $${\hat{U}}_{n+1}$$ U ^ n + 1 is sampled uniformly from $${\hat{U}}_1, \ldots , {\hat{U}}_n$$ U ^ 1 , … , U ^ n , and with complementary probability $$1-p$$ 1 - p , $${\hat{U}}_{n+1}$$ U ^ n + 1 is a new independent uniform variable. The Glivenko–Cantelli theorem remains valid for the reinforced empirical measure, but not the Donsker theorem. Specifically, we show that the sequence of empirical processes converges in law to a Brownian bridge only up to a constant factor when $$p<1/2$$ p < 1 / 2 , and that a further rescaling is needed when $$p>1/2$$ p > 1 / 2 and the limit is then a bridge with exchangeable increments and discontinuous paths. This is related to earlier limit theorems for correlated Bernoulli processes, the so-called elephant random walk, and more generally step reinforced random walks.

Performance Analysis in Heterogeneous Networks with Spatiotemporal Traffic and Scheduling

In this work, we consider interference performance under direct data transmission in a heterogeneous network. The heterogeneous network consists of K-tier base stations and users, whose locations follow independent Poisson point processes (PPPs). Packet arrivals of users follow independent Bernoulli processes. Two different scheduling policies, round-robin (RR) and random scheduling (RS), are employed to all the Base Stations (BS). The universal frequency reuse mode is adopted to reveal actual spectrum reuse. By leveraging stochastic geometry and queueing theory, the interference interactions of the proposed network are accurately modelled. Accurate expressions for the mean packet throughputs of the network under universal frequency reuse mode are derived. The simulation results explore the optical bias factors in heterogeneous networks to maximize the mean packet throughput. Under a given user density, by changing BS densities, we achieved a certain mean packet throughput level.

Spike-Based Winner-Take-All Computation: Fundamental Limits and Order-Optimal Circuits

Winner-take-all (WTA) refers to the neural operation that selects a (typically small) group of neurons from a large neuron pool. It is conjectured to underlie many of the brain's fundamental computational abilities. However, not much is known about the robustness of a spike-based WTA network to the inherent randomness of the input spike trains. In this work, we consider a spike-based [Formula: see text]–WTA model wherein [Formula: see text] randomly generated input spike trains compete with each other based on their underlying firing rates and [Formula: see text] winners are supposed to be selected. We slot the time evenly with each time slot of length 1 ms and model the [Formula: see text] input spike trains as [Formula: see text] independent Bernoulli processes. We analytically characterize the minimum waiting time needed so that a target minimax decision accuracy (success probability) can be reached. We first derive an information-theoretic lower bound on the waiting time. We show that to guarantee a (minimax) decision error [Formula: see text] (where [Formula: see text]), the waiting time of any WTA circuit is at least [Formula: see text]where [Formula: see text] is a finite set of rates and [Formula: see text] is a difficulty parameter of a WTA task with respect to set [Formula: see text] for independent input spike trains. Additionally, [Formula: see text] is independent of [Formula: see text], [Formula: see text], and [Formula: see text]. We then design a simple WTA circuit whose waiting time is [Formula: see text]provided that the local memory of each output neuron is sufficiently long. It turns out that for any fixed [Formula: see text], this decision time is order-optimal (i.e., it matches the above lower bound up to a multiplicative constant factor) in terms of its scaling in [Formula: see text], [Formula: see text], and [Formula: see text].

On a Geometric Extension of the Notion of Exchangeability Referring to Random Events

The notion of exchangeability referring to random events is investigated by using a geometric scheme of representation of possible alternatives. When we distribute among them our sensations of probability, we point out the multilinear essence of exchangeability by means of this scheme. Since we observe a natural one-to-one correspondence between multilinear maps and linear maps, we are able to underline that linearity concept is the most meaningful mathematical concept of probability theory. Exchangeability hypothesis is maintained for mixtures of Bernoulli processes in the same way. We are the first in the world to do this kind of work and for this reason we believe that it is inevitable that our references limit themselves only to those pioneering works which do not keep the real and deep meaning of probability concept a secret, unlike the current ones.