Lecture

Markov Chains: Convergence and Equilibrium

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

Description

This lecture covers the convergence properties of Markov chains, focusing on the Law of Large Numbers and the equilibrium distribution. It explains the classical and Markov Laws of Large Numbers, the frequency of state visits, and the proof of the Markov Law of Large Numbers. The lecture also discusses the asymptotic behavior of Markov chains, the strong Markov property, and the computation of long-run mean rewards. Additionally, it explores the Perron-Frobenius theorem, periodicity, and the rate of convergence in Markov chains.

Official source

Related concepts (32)

Proof theory

Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

Rate of convergence

In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. A sequence that converges to is said to have order of convergence and rate of convergence if The rate of convergence is also called the asymptotic error constant. Note that this terminology is not standardized and some authors will use rate where this article uses order (e.g., ).

Markov chain

A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs now." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC).

Law of large numbers

In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value and tends to become closer to the expected value as more trials are performed. The LLN is important because it guarantees stable long-term results for the averages of some random events.

Proof assistant

In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer. A recent effort within this field is making these tools use artificial intelligence to automate the formalization of ordinary mathematics.

Related lectures (64)

Markov Chain Convergence

Explores Markov chain convergence, emphasizing invariant distribution, Law of Large Numbers, and mean rewards computation.

The Law of Large Numbers: Proof and Applications

Explores the proof and applications of the law of large numbers, emphasizing convergence of the empirical distribution.

Conditional Expectation: Grouping Lemma

Explores conditional expectation, the grouping lemma, and the law of large numbers.

Stochastic Simulations: Ergodicity and Estimators

Explores geometric ergodicity in Markov chains and estimators' bias and variance, highlighting efficiency loss quantification.

Stochastic Simulation: Theory of Markov Chains