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A Note on Exact State Visit Probabilities in Two-State Markov Chains

Mohammad Taha Shah

TL;DR

The paper addresses the problem of determining the exact distribution of the number of visits to a given state in a two-state Markov chain after $N$ transitions. It derives a closed-form expression for $\mathbb{P}(N_l=k\mid N)$ by conditioning on the initial state and partitioning into cases where the final path ends in $S_0$ or $S_1$, using combinatorial counts based on weak compositions. The main contributions are explicit sum expressions for $\mathbb{P}_1(k,N\mid S_1)$ and $\mathbb{P}_2(k,N\mid S_0)$, treatment of boundary cases $k=0$ and $k=N$, and numerical validation. This work enables exact enumeration of state-transition sequences and has implications for queuing, statistical physics, and reinforcement learning, with suggested extensions to multi-state chains.

Abstract

In this note we derive the exact probability that a specific state in a two-state Markov chain is visited exactly $k$ times after $N$ transitions. We provide a closed-form solution for $\mathbb{P}(N_l = k \mid N)$, considering initial state probabilities and transition dynamics. The solution corrects and extends prior incomplete results, offering a rigorous framework for enumerating state transitions. Numerical simulations validate the derived expressions, demonstrating their applicability in stochastic modeling.

A Note on Exact State Visit Probabilities in Two-State Markov Chains

TL;DR

The paper addresses the problem of determining the exact distribution of the number of visits to a given state in a two-state Markov chain after transitions. It derives a closed-form expression for by conditioning on the initial state and partitioning into cases where the final path ends in or , using combinatorial counts based on weak compositions. The main contributions are explicit sum expressions for and , treatment of boundary cases and , and numerical validation. This work enables exact enumeration of state-transition sequences and has implications for queuing, statistical physics, and reinforcement learning, with suggested extensions to multi-state chains.

Abstract

In this note we derive the exact probability that a specific state in a two-state Markov chain is visited exactly times after transitions. We provide a closed-form solution for , considering initial state probabilities and transition dynamics. The solution corrects and extends prior incomplete results, offering a rigorous framework for enumerating state transitions. Numerical simulations validate the derived expressions, demonstrating their applicability in stochastic modeling.

Paper Structure

This paper contains 4 sections, 1 theorem, 4 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Example
  4. Conclusion

Key Result

Theorem 1

Let $S_0$ and $S_1$ be the two states of a Markov chain, with transition probabilities $p_{00}$, $p_{01}$, $p_{10}$, and $p_{11}$, and initial state probabilities $p_0$ and $p_1$ for $S_0$ and $S_1$, respectively. The probability that state $S_1$ is visited exactly $k$ times after $N$ transitions is where $\mathbb{P}_1 (k, N \mid S_1)$ and $\mathbb{P}_2 (k, N \mid S_0)$ are defined as follows: He

Figures (2)

  • Figure 1: State diagram of a two-state Markov chain.
  • Figure 2: All the possible state transition paths considering the initial state is $S_1$ and final state is $S_0$.

Theorems & Definitions (1)

  • Theorem 1