Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Probability and Statistics → Stochastic Processes ↓


Markov Chains


A Markov chain is a mathematical system that experiences transitions from one state to another according to certain probabilistic rules. The unique feature of a Markov chain is that the future state depends only on the current state, not on the sequence of events before it.

Introduction to Markov chains

Consider a simple example like the weather: if you know today's weather, you can predict tomorrow's weather, without needing the weather the day after tomorrow. This is a classic example of the Markov property, which states that the system is "memoryless". Markov chains essentially model situations where this property is true.

Basic concepts

Before delving deeper into Markov chains, let's define some basic terms:

  • State: A description of the state of the Markov system at any time.
  • Transition: The process of moving from one state to another in a Markov chain.
  • Transition Probability: The probability of moving from one state to another.

Mathematical representation

A Markov chain is represented by a matrix called the transition matrix. Each element in the matrix represents the probability of moving from one state to another. Let's look at an example:

P = | 0.7 0.3 | | 0.4 0.6 |

In this matrix, 0.7 is the probability of staying in state 1, and 0.3 is the probability of moving from state 1 to state 2.

Visual example

The following diagram shows a simple Markov chain with two states:

S1 S2 0.3 0.4

In this example, the arrow going from S1 to S2 has a transition probability of 0.3, which means it has a 30% chance of going from state 1 to state 2.

Types of Markov chains

There are mainly two types of Markov chains:

  • Discrete-time Markov chain: This process proceeds in discrete steps or states.
  • Continuous-time Markov chain: Changes (transitions) can happen at any time.

Stable distribution

The Markov chain can reach a point where the probabilities of being in each state become stable. This is called the stationary distribution. It can be calculated by solving the following equation:

πP = π

Here, π denotes the stationary distribution vector, and P is the transition matrix.

Text example

Suppose you are a machine learning engineer working on a recommendation system for a streaming platform. You come across two states that represent user actions: “watching” and “browsing.” The user can switch between these states based on certain probabilities. Using a transition matrix, you represent these probabilities as follows:

P = | 0.8 0.2 | | 0.5 0.5 |

After several iterations, you find that the probability distribution does not change. This stable distribution helps you understand user behavior over time.

Applications of Markov chains

Markov chains are used in a variety of fields, including:

  • Economics: Modeling stock prices and economic trends.
  • Biology: Simulating population dynamics.
  • Queueing theory: modeling systems with service lines.
  • Computational algorithms: such as Google's PageRank algorithm.

Examples in queueing theory

Imagine a single-server queue where customers arrive randomly. The system can be modeled using a Markov chain, where states represent the number of customers in the system. Transition probabilities are based on arrival and service rates.

For example, consider a supermarket with a checkout counter. Customers arrive at random intervals, and the time taken to serve each customer varies. The arrival and service times form the basis of transition probabilities, and the queue length represents the different states of the Markov chain.

Solving Markov chains

Solving a Markov chain typically involves finding the stationary distribution. This is usually done in one of two ways:

  • Iteration: Multiply the initial distribution by the transition matrix until it becomes stationary.
  • Direct solution: Use linear algebra to solve πP = π , with additional constraints such as π sums to 1.

Example calculation

Let's solve the Markov chain using recursion. Suppose we have an initial state distribution: [1, 0]. With a transition matrix:

P = | 0.9 0.1 | | 0.2 0.8 |

Our initial distribution says that we start in state 1 with certainty. After a transition, the distribution becomes:

[1, 0] * | 0.9 0.1 | = | 0.9 0.1 |

Repeating these multiplications will eventually give the stationary distribution.

Conclusion

Markov chains are powerful tools in probability and statistics for modeling random processes. They are particularly useful when the system exhibits Markov properties, where the next state depends only on the current state and therefore it summarizes everything you need to know about the past. Understanding the basics of Markov chains will help you tackle a wide variety of problems in science and engineering.

The topics under Markov chains, such as types, applications, and solutions, are very broad, and continued study can lay a strong foundation for advanced topics in stochastic processes.


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Markov Chains

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