Martingale Theory Research Articles

Noncommutative Nest Hardy Spaces and Martingales

Abstract Let $(\mathcal{M},\tau )$ be a finite von Neumann algebra equipped with a normalized faithful trace and let $\mathbb{A}$ be a nest of order type $\mathbb{N}$. The nest algebra is defined by $H_{\infty }^{r}(\mathbb{A})=\{x\in \mathcal{M}:ex=exe,e\in \mathbb{A}\}$, and the related noncommutative Hardy space $H_{p}^{r}(\mathbb{A})$ is the completion of $H_{\infty }^{r}(\mathbb{A})$ in $L_{p}(\mathcal{M})$. In the present paper, we make a connection between $H_{p}^{r}(\mathbb{A})$ and certain noncommutative martingale Hardy space via showing that triangular truncation is a concrete martingale transform. We include several related results for noncommutative nest Hardy spaces in the spirit of recent development of noncommutative martingale theory.

Capacity and delay performance analysis for large-scale UAV-enabled wireless networks

In this paper, we analyze the capacity and delay performance of a large-scale Unmanned Aerial Vehicle (UAV)-enabled wireless network, in which untethered and tethered UAVs deployed with content files move along with mobile Ground Users (GUs) to satisfy their coverage and content delivery requests. We consider the case where the untethered UAVs are of limited storage, while the tethered UAVs serve as the cloud when the GUs cannot obtain the required files from the untethered UAVs. We adopt the Ornstein-Uhlenbeck (OU) process to capture the mobility pattern of the UAVs moving along the GUs and derive the compact expressions of the coverage probability and capacity of our considered network. Using tools from martingale theory, we model the traffic at UAVs as a two-tier queueing system. Based on this modeling, we further derive the analytical expressions of the network backlog and delay bounds. Through numerical results, we verify the correctness of our analysis and demonstrate how the capacity and delay performance can be significantly improved by optimizing the percentage of the untethered UAVs with cached contents.

A novel and effective method for characterizing time series correlations based on martingale difference correlation.

Analysis of correlation between time series is an essential step for complex system studies and dynamical characteristics extractions. Martingale difference correlation (MDC) theory is mainly concerned with the correlation of conditional mean values between response variables and predictor variables. It is the generalization and deepening of the Pearson correlation coefficient, Spearman correlation coefficient, Kendall correlation coefficient, and other statistics. In this paper, on the basis of phase space reconstruction, the generalized dependence index (GDI) is proposed by using MDC and martingale difference divergence matrix theories, which can measure the degree of dependence between time series more effectively. Moreover, motivated by the theoretical framework of the refined distance correlation method, the corresponding dependence measure (DE) is employed in this paper to construct the DE-GDI plane, so as to comprehensively and intuitively distinguish different types of data and deeply explore the operating mechanism behind the relevant time series and complex systems. According to the performances tested by the different simulated and real-world data, our proposed method performs relatively reasonably and reliably in dependence measuring and data distinguishing. The proposal of this complex data clustering method can not only recognize the features of complex systems but also distinguish them effectively so as to acquire more relevant detailed information.

End-to-End Delay Analysis for Multi-Hop Wireless Links Based on Martingale Theory

End-to-End Delay Analysis for Multi-Hop Wireless Links Based on Martingale Theory

Multi-Agent DRL for task offloading with delay reliability assurances in ris-aided MEC in UDN

Abstract This paper focuses on the scenario where Reconfigurable Intelligent Surfaces (RISs) are introduced in ultra-dense networks (UDN) to guarantee the transmission performance of task offloading of users in Mobile Edge Computing (MEC). Considering users’ task arrival and wireless channel stochasticity, we analyze the delay violation probability bound for each user’s two-hop tandem queueing system with the Martingale theory, i.e., the task offloading and edge computing queues. By taking the delay violation probability bounds as the service reliability constraint, we further propose a MADRL-based algorithm, which co-trains each user and BS as independent and heterogeneous agents and maximizes the total energy efficiency of the system by jointly optimizing the users’ offloading ratio, the RISs’ phase-shift, and the BSs’ resource allocation policy. Experimental results show that our algorithm can improve the EE by about 17.9% compared to other alternative methods under different reliability requirements.

From law of the iterated logarithm to Zolotarev distance for supercritical branching processes in random environment

Consider (Zn)n⩾0 a supercritical branching process in an independent and identically distributed environment. Based on some recent development in martingale limit theory, we established law of the iterated logarithm, strong law of large numbers, invariance principle and optimal convergence rate in the central limit theorem under Zolotarev and Wasserstein distances of order p∈(0,2] for the process (logZn)n⩾0.

Generalized kinetic theory of coarse-grained systems. I. Partial equilibrium and Markov approximations

The general kinetic theory of coarse-grained systems is presented in the abstract formalism of communication theory developed by Shannon and Weaver, Khinchin and Kolmogorov. The martingale theory shows that, under reasonable, general hypotheses, coarse-grained systems can be approximated by generalized Markov systems. For mixing systems, the Kolmogorov entropy production can be defined for nonstationary processes as Kolmogorov defined it for stationary processes.

Detection of Electromagnetic Seismic Precursors from Swarm Data by Enhanced Martingale Analytics.

The detection of seismic activity precursors as part of an alarm system will provide opportunities for minimization of the social and economic impact caused by earthquakes. It has long been envisaged, and a growing body of empirical evidence suggests that the Earth's electromagnetic field could contain precursors to seismic events. The ability to capture and monitor electromagnetic field activity has increased in the past years as more sensors and methodologies emerge. Missions such as Swarm have enabled researchers to access near-continuous observations of electromagnetic activity at second intervals, allowing for more detailed studies on weather and earthquakes. In this paper, we present an approach designed to detect anomalies in electromagnetic field data from Swarm satellites. This works towards developing a continuous and effective monitoring system of seismic activities based on SWARM measurements. We develop an enhanced form of a probabilistic model based on the Martingale theories that allow for testing the null hypothesis to indicate abnormal changes in electromagnetic field activity. We evaluate this enhanced approach in two experiments. Firstly, we perform a quantitative comparison on well-understood and popular benchmark datasets alongside the conventional approach. We find that the enhanced version produces more accurate anomaly detection overall. Secondly, we use three case studies of seismic activity (namely, earthquakes in Mexico, Greece, and Croatia) to assess our approach and the results show that our method can detect anomalous phenomena in the electromagnetic data.

A study on asset price bubble dynamics: explosive trend or quadratic variation?

This paper posits that when an asset exhibits a bubble, its price process can be unbounded from above in finite time with positive probability if a quadratic variation (QV) risk premium is large enough. Based on the local martingale theory of bubbles, we provide sufficient conditions under which price explosion occurs via the QV channel provided that bubbles are present. This QV channel of price explosions is new to the literature and distinct from the explosive autoregressive (AR) dynamics, which is often identified with the presence of bubbles as defined in the time series literature. Using the S&P 500 index and a sample of individual stocks over 1996–2021, we document the existence of price explosions during periods when bubbles occur. Almost all price explosion episodes discovered are associated with the QV and not AR drift channel.

Depths in random recursive metric spaces

Abstract As a generalization of random recursive trees and preferential attachment trees, we consider random recursive metric spaces. These spaces are constructed from random blocks, each a metric space equipped with a probability measure, containing a labelled point called a hook, and assigned a weight. Random recursive metric spaces are equipped with a probability measure made up of a weighted sum of the probability measures assigned to its constituent blocks. At each step in the growth of a random recursive metric space, a point called a latch is chosen at random according to the equipped probability measure, and a new block is chosen at random and attached to the space by joining together the latch and the hook of the block. We use martingale theory to prove a law of large numbers and a central limit theorem for the insertion depth, the distance from the master hook to the latch chosen. We also apply our results to further generalizations of random trees, hooking networks, and continuous spaces constructed from line segments.

Thermodynamics of Computations with Absolute Irreversibility, Unidirectional Transitions, and Stochastic Computation Times

Developing a thermodynamic theory of computation is a challenging task at the interface of nonequilibrium thermodynamics and computer science. In particular, this task requires dealing with difficulties such as stochastic halting times, unidirectional (possibly deterministic) transitions, and restricted initial conditions, features common in real-world computers. Here, we present a framework which tackles all such difficulties by extending the martingale theory of nonequilibrium thermodynamics to generic nonstationary Markovian processes, including those with broken detailed balance and/or absolute irreversibility. We derive several universal fluctuation relations and second-law-like inequalities that provide both lower and upper bounds on the intrinsic dissipation (mismatch cost) associated with any periodic process—in particular, the periodic processes underlying all current digital computation. Crucially, these bounds apply even if the process has stochastic stopping times, as it does in many computational machines. We illustrate our results with exhaustive numerical simulations of deterministic finite automata processing bit strings, one of the fundamental models of computation from theoretical computer science. We also provide universal equalities and inequalities for the acceptance probability of words of a given length by a deterministic finite automaton in terms of thermodynamic quantities, and outline connections between computer science and stochastic resetting. Our results, while motivated from the computational context, are applicable far more broadly. Published by the American Physical Society 2024

Online Stochastic DCA With Applications to Principal Component Analysis.

Stochastic algorithms are well-known for their performance in the era of big data. In this article, we study nonsmooth stochastic Difference-of-Convex functions (DC) programs-the major class of nonconvex stochastic optimization, which have a variety of applications in divers domains, in particular, machine learning. We propose new online stochastic algorithms based on the state-of-the-art DC Algorithm (DCA)-a powerful approach in nonconvex programming framework, in the online context of streaming data continuously generated by some (unknown) source distribution. The new schemes use the stochastic approximations (SAs) principle: deterministic quantities of the standard DCA are replaced by their noisy estimators constructed using newly arriving samples. The convergence analysis of the proposed algorithms is studied intensively with the help of tools from modern convex analysis and martingale theory. Finally, we study several aspects of the proposed algorithms on an important problem in machine learning: the expected problem in principal component analysis (PCA).

Convergence of the Distributed SG Algorithm Under Cooperative Excitation Condition.

In this article, a distributed stochastic gradient (SG) algorithm is proposed where the estimators are aimed to collectively estimate an unknown time-invariant parameter from a set of noisy measurements obtained by distributed sensors. The proposed distributed SG algorithm combines the consensus strategy of the estimation of neighbors with the diffusion of regression vectors. For the theoretical investigation of the proposed algorithm, the main challenge lies in analyzing the influence of the Laplacian matrix on the state transition matrix and the properties of the product of nonindependent and nonstationary random matrices. Some analysis techniques such as graph theory and martingale theory are used to deal with the above issues. A cooperative excitation condition is introduced, under which the convergence of the distributed SG algorithm can be obtained without relying on the independency or stationarity assumptions of regression vectors which are commonly used in the existing literature. Furthermore, the convergence rate of the algorithm can be established. Finally, we show that all the sensors can cooperate to fulfill the estimation task even though any individual sensor cannot by a simulation example.

The Latency Performance Analysis and Effective Relay Selection for Visible Light Networks.

In visible light communication (VLC), the precise latency evaluation of wireless access networks and the efficient forwarding strategy of core networks are the crux for end-to-end reliability provisioning. Leveraging martingale theory, an elegant latency-bounded reliability analysis framework is studied for the VLC network. Considering the characteristic that VLC links are easy to block, the martingale of the service process is constructed. Based on the time shift features, the martingale process related to latency is proposed for the VLC system, which models the influence of entanglement between aggregate arrivals and random service on latency. A stopping time event about latency is defined. Renting the stopping time theory, a tight upper bound of the unreliability with regard to latency is derived. In the core network, a dynamic forward backhaul framework is proposed, which uses the relay selection algorithm based on back-pressure theory to improve data transmission quality. The theoretical latency-bounded reliability matches the simulation results well, which verifies the effectiveness of the proposed analysis framework, and the proposed relay selection algorithm can also improve network performance under data-intensive transmission.

A Hahn-Jordan decomposition and Riesz-Frechet representation theorem in Riesz spaces

We give a Hahn-Jordan decomposition in Riesz spaces which generalizes that of [B.A. Watson, An Andô-Douglas type theorem in Riesz spaces with a conditional expectation, Positivity, 13 (2009), 543–558] and a Riesz-Frechet representation theorem for the T-strong dual, where T is a Riesz space conditional expectation operator. The result of Watson was formulated specifically to assist in the proof of the existence of Riesz space conditional expectation operators with given range space, i.e., a result of Andô-Douglas type. This was needed in the study of Markov processes and martingale theory in Riesz spaces. In the current work, our interest is a Riesz-Frechet representation theorem, for which another variant of the Hahn-Jordan decomposition is required.

Criteria for what makes a local optional martingale a true martingale

What makes an optional stochastic exponential a true optional martingale in a probability space where the underlying filtration is not right continuous nor complete. In this paper, we are going to answer this fundamental question. True martingales play an important role in many applications, just consider the large field of the fair pricing of derivative contracts. Therefore, it is important to give a natural, simple and verifiable condition for having a true martingale property in the context of the theory of optional martingales. The way we have done this here is by extending the method of Beneš to local optional martingales, giving a list of conditions for when an optional stochastic exponential of a local optional martingale is a true optional martingale.

Martingale defects in the volatility surface and bubble conditions in the underlying

AbstractThe martingale theory of bubbles enables testing for asset price bubbles by analyzing option prices. As recently shown by Piiroinen et al. (Asset price bubbles: an option-based indicator, 2018), the SABR model is a strict local martingale when its parameterization implies a positive correlation between stock and option prices. We operationalize this theoretical result and analyze stock price bubbles in 2576 stocks over 26 years. Martingale defect conditions are absorbed quickly by options markets, but identify high proportions in significant and permanent changes in distribution of price returns, option trading activity, short interest in the underlying, and institutional ownership. These results confirm many common assumptions about stock price bubbles. These bubbles are temporally clustered, and tend to occur in periods of positive market development. Martingale defects are rare in market corrections, which indicates that they are a result of overoptimistic speculation.

Maximum Throughput Analysis in Hybrid Energy Harvesting Wireless Communication Systems Based on Martingale Theory

Maximum Throughput Analysis in Hybrid Energy Harvesting Wireless Communication Systems Based on Martingale Theory

Martingale Theory Based Definition and Analysis of Energy Self-Sustainability in Batteryless Internet of Things

Martingale Theory Based Definition and Analysis of Energy Self-Sustainability in Batteryless Internet of Things

WEIGHTED VARIABLE EXPONENT LEBESGUE SPACES ON A PROBABILITY SPACE

In this paper, we introduce the weighted variable exponent Lebesgue spaces defined on a probability space and give some information about the martingale theory of these spaces. We first prove several basic inequalities for expectation operators and obtain several norm convergence conditions for martingales in weighted variable exponent Lebesgue spaces. We discuss the Hölder inequality and embedding properties in these spaces. Finally, under some conditions we investigate Doob’s maximal function.