Question

If \(j\) is a transient state prove that for all \(i\) $$ \sum_{n=1}^{\infty} P_{i j}^{n}<\infty $$ uint: Use relation (5.10).

Short answer

Using the definition of a transient state and relation (5.10), we can derive the sum of the transition probabilities from state \(i\) to state \(j\) as: $$ \sum_{n=1}^{\infty} P_{ij}^{n}= \sum_{n=1}^{\infty} \Bigg(\sum_{\text{all paths between }i \text{ and } j} \prod_{k=1}^{n-1} P_{m_k m_{k+1}}\Bigg). $$ Since state \(j\) is transient, we know that the return probability \(P_{jj}^{(\infty)} \) is less than 1, so the sum of transition probabilities from state \(j\) to itself over all possible \(n\) is also less than 1: $$ \sum_{n=1}^{\infty} P_{jj}^{n} < 1. $$ Then, we have: $$ \sum_{n=1}^{\infty} P_{ij}^{n}< \sum_{n=1}^{\infty} 1 = 1. $$ Hence, for any transient state \(j\), the sum of the transition probabilities from any state \(i\) to state \(j\) over all possible \(n\) is finite: $$ \sum_{n=1}^{\infty} P_{ij}^{n}<\infty. $$

Step-by-step solution

Definition of Transient State

A state \(j\) is called a transient state if there is a positive probability that the process will never return to state \(j\) after leaving it. Mathematically, a state \(j\) is transient if the return probability is less than 1: $$ P_{jj}^{(\infty)}=\sum_{n=1}^{\infty} P_{jj}^{n} < 1. \]

Relation (5.10)

According to relation (5.10), the transition probability \(P_{ij}^{n}\) can be expressed as a sum of products of multiple transition probabilities: $$ P_{ij}^{n} = \sum_{\text{all paths between }i \text{ and } j} \prod_{k=1}^{n-1} P_{m_k m_{k+1}} \] where the sum is taken over all possible paths of length \(n\) between state \(i\) and state \(j\), and \(P_{m_k m_{k+1}}\) are the transition probabilities along that path.

Deriving Sum of Transition Probabilities

We want to find the sum of the transition probabilities from state \(i\) to the transient state \(j\) for all \(n\), i.e., $$ \sum_{n=1}^{\infty} P_{ij}^{n} $$ Applying relation (5.10), we have: $$ \sum_{n=1}^{\infty} P_{ij}^{n}= \sum_{n=1}^{\infty} \sum_{\text{all paths between }i \text{ and } j} \prod_{k=1}^{n-1} P_{m_k m_{k+1}} $$

Transition Probabilities for Transient States

As state \(j\) is a transient state, we know that the return probability \(P_{jj}^{(\infty)}\) is less than 1. Because there is a chance that we never return to state \(j\) again after leaving it, we can also say that the sum of transition probabilities from state \(j\) to itself over all possible \(n\) is also less than 1: $$ \sum_{n=1}^{\infty} P_{jj}^{n} < 1 $$ Then, we can rewrite the expression for the sum of transition probabilities from state \(i\) to state \(j\) as: $$ \sum_{n=1}^{\infty} P_{ij}^{n}= \sum_{n=1}^{\infty} \Bigg(\sum_{\text{all paths between }i \text{ and } j} \prod_{k=1}^{n-1} P_{m_k m_{k+1}}\Bigg) < \sum_{n=1}^{\infty} 1 = 1 $$ This inequality is true because the sum of probabilities is less than 1 for transient states. Thus, we have proven that for \(j\) being a transient state, the sum of the transition probabilities from any state \(i\) to state \(j\) over all possible \(n\) is finite: $$ \sum_{n=1}^{\infty} P_{ij}^{n}<\infty $$

Key concepts

Transition Probabilities

In the realm of stochastic processes, understanding transition probabilities is foundational. These probabilities represent the likelihood of shifting from one state to another in a particular step, or over a number of steps, within a process.

Consider a simple example with dice: if you're moving along a game board based on dice rolls, the chance that you move from your current square (state i) to the next one (state j) is determined by the outcome probabilities of the roll. In mathematical terms, if you are interested in the transition probability after three rolls (three steps), you would look for P_{ij}^3.

For transient states in particular, the total transition probability from state i to state j across an infinite number of steps must be finite—indicating that there is diminishing likelihood of returning to state j as time goes on.

Stochastic Processes

A stochastic process is essentially a collection of random variables representing a process that evolves over time. These variables can take on various states within a given set, and the process describes how these states change.

Consider weather patterns as an example. If today is sunny, what are the odds of rain tomorrow? Weather is a stochastic process with tomorrow's forecast (state) depending on what's happening today (current state) alongside inherently unpredictable elements.

When analyzing stochastic processes, especially more controlled ones like a board game or a computer network's operation, you're dealing with probabilities that shape how states transition over time. Understanding these probabilities helps predict future events and inform decision-making under uncertainty.

State Probability

The state probability refers to the likelihood that a system in a stochastic process is in a particular state at a certain step. In simple terms, it answers questions like: 'What's the chance that the system is in state i at step n?'

To illustrate, imagine a server that can be in one of three states: idle, busy, or down. The state probabilities tell us the likelihood of the server being in any of these states at any given time.

The concept of state probability is closely tied to both transient and steady states within a system. For a transient state, like those discussed in our exercise, the probabilities are not fixed and will vary with time, typically trending towards zero as the number of steps approaches infinity, denoting the system's inherent changes over time.

Markov Chains

Delving into Markov chains unlocks a world where transitional behaviors of stochastic processes have an intriguing property: The future is independent of the past, given the present. This unique attribute means that the next state is only dependent on the current state and not on how the process arrived there.

Visualize a game of snakes and ladders. Every time you roll the dice, your chance of advancing to a specific square does not depend on your prior rolls but only on where you are on the board. In the language of Markov chains, the board positions are the 'states,' and the dice rolls dictate the transitions.

Markov chains are powerful models for a wide range of processes, from predicting stock market trends to modeling user behavior on websites. They are particularly effective when the past is less relevant, allowing predictions and analysis to be more straightforward and computationally efficient.