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Large deviations for template matching between point processes. (English) Zbl 1068.60035

Assume \(X\) is a stationary ergodic process and \(Y\) is a homogeneous Poisson point process. Let \(d\) be the Euclidean distance and denote \( X_a^b = X \cup [a,b)\). For \(t>0\), \(X_0^l \neq 0 \) and \(f\) nonincreasing define \(\rho_l(X_0^l, Y_t^{t+l}) = 1/l \sum_{y \in Y_t^{t+l}} f(d(y-t,X_0^l))\) and \(W_l(\theta, X, Y) = \inf \{t\geq 0 : \rho_l(X_0^l, Y_t^{t+l}) \geq \theta \}.\)
The author proves the large deviation principle for the waiting time \(W_l\) as \(l\) increases, in both cases: \(f\) is scalar and \(f\) is vector-valued. Furthermore, he gives a strong approximation for \(-\log P(W_l(\theta) =0)\). In the case where a certain mixing condition is met, the process \(X\), properly normalized, satisfies a central limit theorem and an a.s. invariance principle. The variance of the normal distribution is given for the case where \(X\) is a homogeneous Poisson process as well.

MSC:

60F10 Large deviations
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)

Keywords:

waiting time
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References:

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