The distribution of the maximum of a first order moving average: the discrete case
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We give the distribution of $M_n$, the maximum of a sequence of $n$ observations from a moving average of order 1. Solutions are first given in terms of repeated integrals and then for the case where the underlying independent random variables are discrete. When the correlation is positive, $$ P(M_n \max^n_{i=1} X_i \leq x) = \sum_{j=1}^\infty β_{jx} ν_{jx}^{n} \approx B_{x} r{1x}^{n} $$ where $\{ν_{jx}\}$ are the eigenvalues of a certain matrix, $r_{1x}$ is the maximum magnitude of the eigenvalues, and $I$ depends on the number of possible values of the underlying random variables. The eigenvalues do not depend on $x$ only on its range.
13 pages. This version gives full solutions to the examples
13 pages. This version gives full solutions to the examples