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1. Testing Symmetric Markov Chains from a Single Trajectory

   Daskalakis, Constantinos; Dikkala, Nishanth; Gravin, Nick (January 2018, 31st Annual Conference on Learning Theory)

   Classical distribution testing assumes access to i.i.d. samples from the distribution that is being tested. We initiate the study of Markov chain testing, assuming access to a single trajectory of a Markov Chain. In particular, we observe a single trajectory X0,...,Xt,... of an unknown, symmetric, and finite state Markov Chain M. We do not control the starting state X0, and we cannot restart the chain. Given our single trajectory, the goal is to test whether M is identical to a model Markov Chain M0 , or far from it under an appropriate notion of difference. We propose a measure of difference between two Markov chains, motivated by the early work of Kazakos [Kaz78], which captures the scaling behavior of the total variation distance between trajectories sampled from the Markov chains as the length of these trajectories grows. We provide efficient testers and information-theoretic lower bounds for testing identity of symmetric Markov chains under our proposed measure of difference, which are tight up to logarithmic factors if the hitting times of the model chain M0 is O(n) in the size of the state space n. 

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2. Testing Symmetric Markov Chains from a Single Trajectory

   Constantinos Daskalakis, Nishanth Dikkala (July 2018, Conference on Learning Theory (COLT))

   The paper’s abstract in valid LaTeX, without non-standard macros or \cite commands. Classical distribution testing assumes access to i.i.d. samples from the distribution that is being tested. We initiate the study of Markov chain testing, assuming access to a {\em single trajectory of a Markov Chain.} In particular, we observe a single trajectory X0,…,Xt,… of an unknown, symmetric, and finite state Markov Chain M. We do not control the starting state X0, and we cannot restart the chain. Given our single trajectory, the goal is to test whether M is identical to a model Markov Chain M′, or far from it under an appropriate notion of difference. We propose a measure of difference between two Markov chains, motivated by the early work of Kazakos [78], which captures the scaling behavior of the total variation distance between trajectories sampled from the Markov chains as the length of these trajectories grows. We provide efficient testers and information-theoretic lower bounds for testing identity of symmetric Markov chains under our proposed measure of difference, which are tight up to logarithmic factors if the hitting times of the model chain M′ is O~(n) in the size of the state space n. 

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3. Testing Symmetric Markov Chains from a Single Trajectory

   Daskalakis, Constantinos; Dikkala, Nishanth; Gravin, Nick (July 2018, Conference on Learning Theory (COLT))

   The paper’s abstract in valid LaTeX, without non-standard macros or \cite commands. Classical distribution testing assumes access to i.i.d. samples from the distribution that is being tested. We initiate the study of Markov chain testing, assuming access to a {\em single trajectory of a Markov Chain.} In particular, we observe a single trajectory X0,…,Xt,… of an unknown, symmetric, and finite state Markov Chain M. We do not control the starting state X0, and we cannot restart the chain. Given our single trajectory, the goal is to test whether M is identical to a model Markov Chain M′, or far from it under an appropriate notion of difference. We propose a measure of difference between two Markov chains, motivated by the early work of Kazakos [78], which captures the scaling behavior of the total variation distance between trajectories sampled from the Markov chains as the length of these trajectories grows. We provide efficient testers and information-theoretic lower bounds for testing identity of symmetric Markov chains under our proposed measure of difference, which are tight up to logarithmic factors if the hitting times of the model chain M′ is O~(n) in the size of the state space n. 

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4. Numerical Solutions for Detecting Contingency in Modern Power Systems

   https://doi.org/10.1109/ICTC57116.2023.10154790  

   Ma, Xiaohang; Qian, Hongjiang; Wang, Le Yi; Nazari, Masoud H.; Yin, George (May 2023, Proc. 2023 4th IEEE Inform. Comm. Tech. Conference)

   This paper is devoted to the detection of contingencies in modern power systems. Because the systems we consider are under the framework of cyber-physical systems, it is necessary to take into consideration of the information processing aspect and communication networks. A consequence is that noise and random disturbances are unavoidable. The detection problem then becomes one known as quickest detection. In contrast to running the detection problem in a discretetime setting leading to a sequence of detection problems, this work focuses on the problem in a continuous-time setup. We treat stochastic differential equation models. One of the distinct features is that the systems are hybrid involving both continuous states and discrete events that coexist and interact. The discrete event process is modeled by a continuous-time Markov chain representing random environments that are not resented by a continuous sample path. The quickest detection then can be written as an optimal stopping problem. This paper is devoted to finding numerical solutions to the underlying problem. We use a Markov chain approximation method to construct the numerical algorithms. Numerical examples are used to demonstrate the performance. 

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5. Uniform-in-time bounds for a stochastic hybrid system with fast periodic sampling and small white-noise

   https://doi.org/10.1142/S0219493725500066  

   Dhama, Shivam Singh; Spiliopoulos, Konstantinos (February 2025, Stochastics and Dynamics)

   We study the asymptotic behavior, uniform-in-time, of a nonlinear dynamical system under the combined effects of fast periodic sampling with period [Formula: see text] and small white noise of size [Formula: see text]. The dynamics depend on both the current and recent measurements of the state, and as such it is not Markovian. Our main results can be interpreted as Law of Large Numbers (LLN) and Central Limit Theorem (CLT) type results. LLN type result shows that the resulting stochastic process is close to an ordinary differential equation (ODE) uniformly in time as [Formula: see text] Further, in regards to CLT, we provide quantitative and uniform-in-time control of the fluctuations process. The interaction of the small parameters provides an additional drift term in the limiting fluctuations, which captures both the sampling and noise effects. As a consequence, we obtain a first-order perturbation expansion of the stochastic process along with time-independent estimates on the remainder. The zeroth- and first-order terms in the expansion are given by an ODE and SDE, respectively. Simulation studies that illustrate and supplement the theoretical results are also provided. 

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