Large deviation results for waiting times in repeated experiments (Q1063927)

Consider an i.i.d. sequence \(Z_ 1,Z_ 2,..\). with a finite set of possible values \(\Sigma\) and let \(A_ i\) \((i=1,...,r)\) be finite sequences of length k over \(\Sigma\). The author presents exact upper and lower exponential bounds for the joint probabilities \(P(\tau_ 1/E\tau_ 1>x_ 1,...,\tau_ r/E\tau_ r>x_ r)\), where \(\tau_ i\) denotes the waiting time until \(A_ i\) occurs as a run in the sequence \(Z_ 1,Z_ 2,...\), and \(x_ 1,...,x_ r\) are arbitrary positive numbers. This is a large deviation type result which, in particular, establishes asymptotic independence and exponentiality of the stopping times \(\tau_ i\).