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Chapter 6: Problem 11

Let \(\left\\{X_{n}: n \geq 1\right)\) be independent identically distributed integer-valued random variables. Let \(S_{n}=\sum_{r=1}^{n} X_{r}\), with \(S_{0}=0, Y_{n}=X_{n}+X_{n-1}\) with \(X_{0}=0\), and \(Z_{n}=\sum_{r=0}^{n} S_{r} .\) Which of the following constitute Markov chains: (a) \(S_{n}, \quad\) (b) \(Y_{n}\), (c) \(Z_{n}-\) (d) the sequence of pairs \(\left(S_{n}, Z_{n}\right) ?\)

Short Answer

Expert verified
Only \(S_n\) and pair \((S_n, Z_n)\) form Markov chains.

Step by step solution

01

Understanding Markov Chains

A stochastic process \(\left\{X_n\right\}\) is a Markov chain if the future state depends only on the present state and not on the past states. Formally, this is \( \mathbb{P}(X_{n+1}=x_{n+1} | X_n=x_n, X_{n-1}=x_{n-1}, \dots, X_1=x_1) = \mathbb{P}(X_{n+1}=x_{n+1} | X_n=x_n) \).
02

Evaluating \( S_n \)

\( S_n = \sum_{r=1}^{n} X_r \) is the sum of independent and identically distributed variables. The change from \(S_n\) to \(S_{n+1}\) depends only on the addition of \(X_{n+1}\) (since \( S_{n+1} = S_n + X_{n+1} \)), thus \( S_n \) forms a Markov chain.
03

Evaluating \( Y_n \)

\( Y_n = X_n + X_{n-1} \) depends on the last two variables \(X_n\) and \(X_{n-1}\). Therefore, the transition to \(Y_{n+1}\) depends on both \(Y_n\) and \(X_{n+1}\), not solely on \(Y_n\). Hence, \( Y_n \) does not form a Markov chain.
04

Evaluating \( Z_n \)

\( Z_n = \sum_{r=0}^{n} S_r \) is a cumulative sum of \(S_r\)s, where each \(S_r\) is the sum of all \(X_i\)s up to \(r\). The dependency on all prior \(X_i\) makes \(Z_n\) non-Markovian as the next state depends on the entire history, not just the current state.
05

Evaluating Pair \((S_n, Z_n)\)

Consider the pair \((S_n, Z_n)\). \(S_n\) itself is a Markov chain, and changes in \(Z_{n+1}\) are determined by \(Z_n + S_{n+1}\). Hence, transitions in \((S_n, Z_n)\) depend only on the current state \((S_n, Z_n)\), making this pair a Markov chain.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Random Variables
Random variables are fundamental concepts in probability and statistics. They are used to represent possible outcomes of a random phenomenon or experiment. Imagine rolling a dice; each face of the dice is a potential outcome, and rolling it is a random process.

In mathematical terms, a random variable is a function that assigns real numbers to outcomes in the sample space of a stochastic experiment. These variables can be discrete, like the number of heads when flipping coins, or continuous, such as measuring heights in a population.

Random variables help in making mathematical models of probabilistic scenarios. They allow us to compute probabilities and expectations using statistical theories and techniques.
Stochastic Processes
Stochastic processes involve a sequence of random variables used to model systems that evolve over time in a random manner. They are crucial in fields such as finance, physics, and engineering to model time-dependent behaviors.

A Markov Chain is a specific type of stochastic process. Its distinct feature is that the future state depends only on the present state and not on the sequence of events that preceded it. This is called the "memoryless" property. For example, predicting the weather tomorrow based only on today's weather, not yesterday's or any day before.

Stochastic processes can be discrete time, where changes occur at specific intervals, or continuous time, where changes can happen at any moment.
iid (independent and identically distributed)
The term 'iid' stands for independent and identically distributed. It refers to a set of random variables with two critical properties. First, *independent* means the occurrence of one event does not affect the other. Second, *identically distributed* indicates all the variables follow the same probability distribution.

Imagine multiple coin flips: each flip doesn鈥檛 affect the others and all have the same probability of heads or tails. In mathematical modeling, this assumption simplifies analyzing complex systems. It implies each variable has the same role and impact, easing the application of statistical methods.

Understanding iid variables is significant in the study of Markov Chains and stochastic processes as it lays the groundwork for complex probabilistic models.
Cumulative Sums
Cumulative sums are the totals accumulated from a sequence of numbers. In the context of random variables, we call the sum of a series of random variables a cumulative sum. It's akin to saving a certain amount of money regularly and keeping track of your total savings.

Mathematically, if you have a sequence of random variables \(X_1, X_2, X_3, \ldots\), the cumulative sum \(S_n\) at step \(n\) is represented by \(S_n = X_1 + X_2 + \ldots + X_n\).

In the provided exercise, \(S_n\) denotes the cumulative sum of iid random variables. Cumulative sums are vital in many areas including finance, quality control, and signal processing. Their study aids in understanding the overall trends and variations over time.

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Most popular questions from this chapter

A die is rolled repeatedly. Which of the following are Markov chains? For those that are, supply the transition matrix. (a) The largest number \(X_{n}\) shown up to the \(n\)th roll. (b) The number \(N_{n}\) of sixes in \(n\) rolls. (c) At time \(r\), the time \(C_{r}\) since the most recent six. (d) At time \(r\), the time \(B_{r}\) until the next six.

A particle has velocity \(V(t)\) at time \(t\), where \(V(t)\) is assumed to take values in \(\left(n+\frac{1}{2} ; n \geq 0\right]\). Transitions during \((t, t+h)\) are possible as follows: $$ P(V(t+h)=w \mid V(t)=v)= \begin{cases}\left(v+\frac{1}{2}\right) h+o(h) & \text { if } w=v+1 \\ 1-2 v h+o(h) & \text { if } w=v \\\ \left(v-\frac{1}{2}\right) h+o(h) & \text { if } w=v-1\end{cases} $$ Initially \(V(0)=\frac{1}{2}\), Let $$ G(s, t)=\sum_{n=0}^{\infty} s^{n} \mathrm{P}\left(V(t)=n+\frac{1}{2}\right) $$ (a) Show that $$ \frac{\partial G}{\partial t}=(1-s)^{2} \frac{\partial G}{\partial s}-(1-s) G $$ and deduce that \(G(s, t)=(1+(1-s) t\\}^{-1}\). (b) Show that the expected length \(m_{n}(T)\) of time for which \(V=n+\frac{1}{2}\) during the time interval \([0, T]\). is given by $$ m_{n}(T)=\int_{0}^{T} \mathrm{P}\left(V(t)=n+\frac{1}{2}\right) d t $$ and that, for fixed \(k, m_{k}(T)-\log T \rightarrow-\sum_{i=1}^{k} i^{-1}\) as \(T \rightarrow \infty\). (c) What is the expected velocity of the particle at time \(t\) ?

Let \(N\) be a birth process with intensities \(\lambda_{0}, \lambda_{1}, \ldots\), and let \(N(0)=0\). Show that \(p_{n}(t)=\) \(P(N(t)=n)\) is given by $$ p_{n}(t)=\frac{1}{\lambda_{n}} \sum_{i=0}^{n} \lambda_{i} e^{-\lambda_{i} t} \prod_{j=0 \atop j \neq i}^{n} \frac{\lambda_{j}}{\lambda_{j}-\lambda_{i}} $$ provided that \(\lambda_{i} \neq \lambda_{j}\) whenever \(i \neq j\).

Jobs arrive in a computer queue in the manner of a Poisson process with intensity \(\lambda\). The central processor handles them one by one in the order of their arrival, and each has an exponentially distributed runtime with parameter \(\mu\), the runtimes of different jobs being independent of each other and of the arrival process. Let \(X(t)\) be the number of jobs in the system (either running or waiting) at time \(t\), where \(X(0)=0 .\) Explain why \(X\) is a Markov chain, and write down its generator. Show that a stationary distribution exists if and only if \(\lambda<\mu\), and find it in this case.

Consider the symmetric random walk in three dimensions on the set of points \((x, y, z): x, y, z=\) \(0, \pm 1, \pm 2, \ldots, 1 ;\) this process is a sequence \(\left[\mathbf{X}_{n}: n \geq 0\right]\) of points such that \(\mathbb{P}\left(\mathbf{X}_{n+1}=\mathbf{X}_{n}+\epsilon\right)=\frac{1}{6}\) for \(\epsilon=(\pm 1,0,0),(0, \pm 1,0),(0,0, \pm 1)\). Suppose that \(\mathbf{X}_{0}=(0,0,0)\). Show that $$ \mathrm{P}\left(\mathbf{X}_{2 n}=(0,0,0)\right)=\left(\frac{1}{6}\right)^{2 n} \sum_{i+j+k=n} \frac{(2 n) !}{(i) j ! k !)^{2}}=\left(\frac{1}{2}\right)^{2 n}\left(\begin{array}{c} 2 n \\ n \end{array}\right) \sum_{i+j+h=n}\left(\frac{n !}{3^{n} i 1 j ! k !}\right)^{2} $$ and deduce by Stirling's formula that the origin is a transient state.
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