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Zengjing Chen

Publications and source records attributed to Zengjing Chen.

At least 19 recordsLinked to original sources
arXiv2026

Deviation Tests for a High-dimensional Mean

This paper investigates testing for deviation of a high-dimensional mean vector $\boldsymbol{\mu}$. In contrast to the standard one-sample significance test of the form: $H_0^\texttt{e} : \boldsymbol{\mu} = \boldsymbol{\mu}_0$ versus $H_1^\texttt{e} : \boldsymbol{\mu} \neq \boldsymbol{\mu}_0$, we focus on testing the deviation $H_0 : \|\boldsymbol{\mu} - \boldsymbol{\mu}_0\|_2 \ge d_0$ versus $H_1 : \|\boldsymbol{\mu} - \boldsymbol{\mu}_0\|_2 < d_0$ for a prespecified length $d_0 > 0$. Constructing a valid test statistic for this problem is technically nontrivial. By applying the concept of positive and negative feedback processes from control theory, we propose a test statistic based on a two-armed bandit (TAB) process. The deviation test is also extended to the two-sample setting. Simulation experiments confirm a good performance of the tests in finite samples. Finally, a real data analysis demonstrates the practical significance of the proposed deviation tests.

stat.ME↗
arXiv2025

Optimization via Strategic Law of Large Numbers

This paper proposes a unified framework for the global optimization of a continuous function in a bounded rectangular domain. Specifically, we show that: (1) under the optimal strategy for a two-armed decision model, the sample mean converges to a global optimizer under the Strategic Law of Large Numbers, and (2) a sign-based strategy built upon the solution of a parabolic PDE is asymptotically optimal. Motivated by this result, we propose a class of {\bf S}trategic {\bf M}onte {\bf C}arlo {\bf O}ptimization (SMCO) algorithms, which uses a simple strategy that makes coordinate-wise two-armed decisions based on the signs of the partial gradient of the original function being optimized over (without the need of solving PDEs). While this simple strategy is not generally optimal, we show that it is sufficient for our SMCO algorithm to converge to local optimizer(s) from a single starting point, and to global optimizers under a growing set of starting points. Numerical studies demonstrate the suitability of our SMCO algorithms for global optimization, and illustrate the promise of our theoretical framework and practical approach. For a wide range of test functions with challenging optimization landscapes (including ReLU neural networks with square and hinge loss), our SMCO algorithms converge to the global maximum accurately and robustly, using only a small set of starting points (at most 100 for dimensions up to 1000) and a small maximum number of iterations (200). In fact, our algorithms outperform many state-of-the-art global optimizers, as well as local algorithms augmented with the same set of starting points as ours.

math.OC↗
arXiv2025

Stable Acoustic Relay Assignment with High Throughput via Lase Chaos-based Reinforcement Learning

This study addresses the problem of stable acoustic relay assignment in an underwater acoustic network. Unlike the objectives of most existing literature, two distinct objectives, namely classical stable arrangement and ambiguous stable arrangement, are considered. To achieve these stable arrangements, a laser chaos-based multi-processing learning (LC-ML) method is introduced to efficiently obtain high throughput and rapidly attain stability. In order to sufficiently explore the relay's decision-making, this method uses random numbers generated by laser chaos to learn the assignment of relays to multiple source nodes. This study finds that the laser chaos-based random number and multi-processing in the exchange process have a positive effect on higher throughput and strong adaptability with environmental changing over time. Meanwhile, ambiguous cognitions result in the stable configuration with less volatility compared to accurate ones. This provides a practical and useful method and can be the basis for relay selection in complex underwater environments.

cs.SD↗
arXiv2024

Proof of a conjecture about Parrondo's paradox for two-armed slot machines

The 1936 Mills Futurity slot machine had the feature that, if a player loses 10 times in a row, the 10 lost coins are returned. Ethier and Lee (2010) studied a generalized version of this machine, with 10 replaced by deterministic parameter J. They established the Parrondo effect for a hypothetical two-armed machine with the Futurity award. Specifically, arm A and arm B, played individually, are asymptotically fair, but when alternated ran-domly (the so-called random mixture strategy), the casino makes money in the long run. They also considered the nonrandom periodic pattern strategy for patterns with r As and s Bs (e.g., ABABB if r = 2 and s = 3). They established the Parrondo effect if r + s divides J, and conjectured it in four other situations, including the case J = 2 with r >= 1 and s >= 1. We prove the conjecture in the latter case.

math.PR↗
arXiv2024

Optimal State Equation for the Control of a Diffusion with Two Distinct Dynamics

We consider a class of stochastic control problems which has been widely used in optimal foraging theory. The state processes have two distinct dynamics, characterized by two pairs of drift and diffusion coefficients, depending on whether it takes values bigger or smaller than a threshold value. Adopting a perturbation type approach, we find an expression for potential measure of the optimal state process. We then obtain an expression for the transition density of the optimal state process by inverting the associated Laplace transform. Properties including the stationary distribution of the optimal state process are discussed. Finally, the expression of the value function is given for this class stochastic control problems.

math.OC↗
arXiv2023

Approximate optimality and the risk/reward tradeoff in a class of bandit problems

This paper studies a sequential decision problem where payoff distributions are known and where the riskiness of payoffs matters. Equivalently, it studies sequential choice from a repeated set of independent lotteries. The decision-maker is assumed to pursue strategies that are approximately optimal for large horizons. By exploiting the tractability afforded by asymptotics, conditions are derived characterizing when specialization in one action or lottery throughout is asymptotically optimal and when optimality requires intertemporal diversification. The key is the constancy or variability of risk attitude. The main technical tool is a new central limit theorem.

econ.TH↗
arXiv2022

Long bet will lose: demystifying seemingly fair gambling via two-armed Futurity bandit

No matter how much some gamblers occasionally win, as long as they continue to gamble, sooner or later they will lose more to the casino, which is the so-called long bet will lose. Our results demonstrate the counter-intuitive phenomenon, that gamblers involved in long bets will lose but casinos always advertise their unprofitable circumstances. Here we expose the law of inevitability behind long bet will loss by theoretically and experimentally demystifying the profitable mystery behind casinos under two-armed antique Mills Futurity slot machine. The main results straightforwardly elucidate that all casino projects are seemingly a fair gamble but essentially unfair, i.e., the casino's win rate is greater than 50%. We anticipate our assay to be a starting point for studying the fairness of more sophisticated multi-armed Futurity bandits based on the mathematical tool. In application, a fairness study of the Futurity bandits not only exposes the fraud of casinos for gamblers but also discloses discount marketing, bundled sales, or other induced consumption tactics.

q-fin.GN↗
arXiv2022

A Confirmation of a Conjecture on the Feldman's Two-armed Bandit Problem

Myopic strategy is one of the most important strategies when studying bandit problems. In this paper, we consider the two-armed bandit problem proposed by Feldman. With general distributions and utility functions, we obtain a necessary and sufficient condition for the optimality of the myopic strategy. As an application, we could solve Nouiehed and Ross's conjecture for Bernoulli two-armed bandit problems that myopic strategy stochastically maximizes the number of wins.

math.ST↗
arXiv2022

A Central Limit Theorem, Loss Aversion and Multi-Armed Bandits

This paper studies a multi-armed bandit problem where the decision-maker is loss averse, in particular she is risk averse in the domain of gains and risk loving in the domain of losses. The focus is on large horizons. Consequences of loss aversion for asymptotic (large horizon) properties are derived in a number of analytical results. The analysis is based on a new central limit theorem for a set of measures under which conditional variances can vary in a largely unstructured history-dependent way subject only to the restriction that they lie in a fixed interval.

math.PR↗
arXiv2022

Strategy-Driven Limit Theorems Associated Bandit Problems

Motivated by the study of asymptotic behaviour of the bandit problems, we obtain several strategy-driven limit theorems including the law of large numbers, the large deviation principle, and the central limit theorem. Different from the classical limit theorems, we develop sampling strategy-driven limit theorems that generate the maximum or minimum average reward. The law of large numbers identifies all possible limits that are achievable under various strategies. The large deviation principle provides the maximum decay probabilities for deviations from the limiting domain. To describe the fluctuations around averages, we obtain strategy-driven central limit theorems under optimal strategies. The limits in these theorem are identified explicitly, and depend heavily on the structure of the events or the integrating functions and strategies. This demonstrates the key signature of the learning structure. Our results can be used to estimate the maximal (minimal) rewards, and to identify the conditions of avoiding the Parrondo's paradox in the two-armed bandit problem. It also lays the theoretical foundation for statistical inference in determining the arm that offers the higher mean reward.

math.PR↗
arXiv2020

A Central Limit Theorem for Sets of Probability Measures

We prove a central limit theorem for a sequence of random variables whose means are ambiguous and vary in an unstructured way. Their joint distribution is described by a set of measures. The limit is (not the normal distribution and is) defined by a backward stochastic differential equation that can be interpreted as modeling an ambiguous continuous-time random walk.

math.PR↗
arXiv2020

Explicit solutions for a class of nonlinear backward stochastic differential equations and their nodal sets

In this paper, we investigate a class of nonlinear backward stochastic differential equations (BSDEs) arising from financial economics, and give specific information about the nodal sets of the related solutions. As applications, we are able to obtain the explicit solutions to an interesting class of nonlinear BSDEs including the k-ignorance BSDE arising from the modeling of ambiguity of asset pricing.

math.PR↗
arXiv2019

Non-uniform Berry-Esseen Bound by Unbounded Exchangeable Pair Approach

In this paper, a new technique is introduced to obtain non-uniform Berry-Esseen bounds of normal and nonnormal approximation for unbounded exchangeable pairs. This technique does not rely on the concentration inequalities developed by Chen and Shao \cite{cls1, cls2} and can be applied to the quadratic forms, general Curie-Weiss model and an independence test. In particular, our non-uniform result about the independence test is under 6th moment condition, while the uniform bound in Chen and Shao \cite{cs2} requires 24th moment condition.

math.ST↗
arXiv2017

Strong laws of large numbers for sub-linear expectations

We investigate three kinds of strong laws of large numbers for capacities with a new notion of independently and identically distributed (IID) random variables for sub-linear expectations initiated by Peng. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov's strong law of large numbers to the case where probability measures are no longer additive. An important feature of these strong laws of large numbers is to provide a frequentist perspective on capacities.

math.PR↗
arXiv2013

A law of the iterated logarithm sublinear expectations

In this paper, motivated by the notion of independent identically distributed (IID) random variables under sub-linear expectations initiated by Peng, we investigate a law of the iterated logarithm for capacities. It turns out that our theorem is a natural extension of the Kolmogorov and the Hartman-Wintner laws of the iterated logarithm.

math.PR↗
arXiv2011

A new comparison theorem of multidimensional BSDEs

In this paper, we define a new total order on R^N and use this order together with backward stochastic viability property(for short BSVP) to study the property of the generator of backward stochastic differential equation(for short BSDE) when the price of contingent claim can be represented by a BSDE in the no-arbitrage financial market. The main result is the necessary and sufficient condition for comparison theorem of multidimensional BSDEs under this order.

math.PR↗
arXiv2011

An Invariance Principle of G-Brownian Motion for the Law of the Iterated Logarithm under G-expectation

The classical law of the iterated logarithm (LIL for short)as fundamental limit theorems in probability theory play an important role in the development of probability theory and its applications. Strassen (1964) extended LIL to large classes of functional random variables, it is well known as the invariance principle for LIL which provide an extremely powerful tool in probability and statistical inference. But recently many phenomena show that the linearity of probability is a limit for applications, for example in finance, statistics. As while a nonlinear expectation--- G-expectation has attracted extensive attentions of mathematicians and economists, more and more people began to study the nature of the G-expectation space. A natural question is: Can the classical invariance principle for LIL be generalized under G-expectation space? This paper gives a positive answer. We present the invariance principle of G-Brownian motion for the law of the iterated logarithm under G-expectation.

math.PR↗