Mathematics > Probability > Markov Chains

Description:

Markov Chains are a fundamental topic within the field of probability, characterized by their utility in modeling systems that transition from one state to another in a probabilistic manner. These systems are governed by the Markov property, which stipulates that the future state of the system depends only on the present state, not on the sequence of states that preceded it. This property is often summarized as “memorylessness.”

More formally, a Markov Chain is a stochastic process {X_n} defined on a state space S, where the probability of transitioning from the current state \(i \in S\) to the next state \(j \in S\) is given by the conditional probability:

\[ P(X_{n+1} = j \mid X_n = i, X_{n-1} = i_{n-1}, \ldots , X_0 = i_0) = P(X_{n+1} = j \mid X_n = i) = P_{ij} \]

This relationship implies that the process is memoryless, and the transitions depend solely on the present state.

Markov Chains can be represented through a transition matrix \( P \), which is an \( |S| \times |S| \) matrix where each entry \( P_{ij} \) represents the probability of transitioning from state \( i \) to state \( j \). The elements of this matrix have the property that:

\[ \sum_{j \in S} P_{ij} = 1 \quad \text{for all} \ i \in S \]

There are various types of Markov Chains based on different properties:

1. Discrete-Time Markov Chains (DTMC): The process moves between states at discrete time intervals.
2. Continuous-Time Markov Chains (CTMC): The process changes states continuously over time.

Equilibrium and Stationary Distributions:
A key concept within Markov Chains is the idea of equilibrium, or stationary distributions. A stationary distribution \( \pi \) is a probability distribution over the states such that once the Markov Chain reaches this distribution, it remains unchanged. Mathematically, \( \pi \) satisfies:

\[ \pi P = \pi \]

This signifies that the distribution \( \pi \) does not change after applying the transition matrix \( P \).

Applications:
Markov Chains have wide-ranging applications in various fields, including:

• Physics: Modeling of particles in dynamic systems.
• Economics and Finance: Stock market analysis, economic forecasting.
• Computer Science: Algorithms for random walks, Markov decision processes in reinforcement learning.
• Biology: Population genetics, spread of diseases modeled through epidemiological processes.

By providing a rigorous mathematical framework, Markov Chains enable the modeling, analysis, and prediction of systems that evolve in probabilistic and temporal dimensions, making them indispensable tools in theoretical and applied research.