I need help finding a theorem which could be used to prove that the Wald-Wolfowitz runs test is asymptotically normal. Let me formalize my question.

We have a random sample $\{X_0,X_1,...,X_n\}$ (if needed those random variable can be assumed to have a continuous density function) and let $\hat m_n$ be the sample median. Then we can define the following r.v.:

$$B_i = \left\{\begin{array}{ccc} 0 & \text{if} & X_i\leq \hat m_n \\ 1 & \text{if} & X_i> \hat m_n \end{array}\right.$$

With those $B_i$'s we define:

$$I_i = \left\{\begin{array}{ccc} 0 & \text{if} & B_i= B_{i-1} \\ 1 & \text{if} & B_i\neq B_{i-1} \end{array}\right.,$$

for $i=1,2,...,n$. With this notation the sum $S_n = 1+ \sum_{i=1}^{n}I_i$ is equal to the number of runs.

Wald and Wolfowitz original paper show that this statistic is asymptotically normal by explicitly finding the limit of $\mathbb P(\frac{S_n-\mu}{\sigma\sqrt{n}}\leq t)$, for appropriate values of $\mu$ and $\sigma$.

Isn't there a way to show the same think with the help of some kind of central limit theorem for dependente variables? Can anyone point me a direction in the literature?


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