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Recall that if \(X_1, X_2, \ldots\) are i.i.d. random variables with finite nonzero variance then if \(S_n = X_1 + \cdots + X_n\), we have that the normalized sum
\[Z_n = \frac{S_n - \mathbb{E} S_n} {\sqrt{ var(S_n) } } = \frac{S_n - n\mathbb{E}X_1} {\sqrt{ n \cdot var(X_1) } }\] converges in distribution to a standard normal.
Illustrate this fact in R by running simulations. Specifically, consider the case where \(X_i\) are i.i.d. exponential random variables with mean \(1\), and for large values of \(n\), simulate \(k\) instances of \(Z_n\), where \(k\) is also large.
Consider the transition matrix \(P\) on three states \(\{1,2,3\}\) given by \[ \begin{equation*} P = \begin{pmatrix} 1/4 & 1/4 & 1/2 \\ 1/4 & 1/4 & 1/2 \\ 1/8 & 1/4 & 5/8 \end{pmatrix} \end{equation*} \]
Let \(X\) be a Markov chain with transition matrix \(P\) as in the previous question. Let \(\pi\) be the stationary distribution and assume that \(X\) is started at stationarity, so that \(X_i\) has law \(\pi\) for all \(i\). Consider the normalized sum given by
\[Z_n = \frac{S_n - \mathbb{E} S_n} {\sqrt{ n}} = \frac{S_n - n\mathbb{E} X_1} {\sqrt{ n}}\] where \(S_n = X_1 + \cdots + X_n.\)
Suppose that \(X\) is an aperiodic and irreducible Markov chain on a finite number of states that is started at the stationary distribution. Suppose that all you see is the realization \((x_0,x_1, \ldots, x_n)\), where \(n\) is large. How can you estimate the values transition matrix \(P\)? Carefully explain your answer and why your estimates are reasonable.
Suppose someone hands you the data for arrival times (in seconds) of visits to some website. It is given by \[ \begin{eqnarray*} && arr = (0.16, 0.28, 0.42, 0.66, 0.75, \\ && 1.27, 2.25, 2.33, 2.83, 4.09, \\ && 4.30, 4.80, 5.20, 5.68, 7.05) \end{eqnarray*} \] Suppose that a Poisson process is a good model for these arrival times. Use this data and the assumption that it is a Poisson process to find estimates for the following. Justify your answers.