Abstract
In many applications it is important to evaluate the impact of clusters of observations caused by the dependence and heaviness of tails in time series. We consider a stationary sequence of random variables {R n } n ≥ 1 with marginal cumulative distribution function F(x) and the extremal index θ ∈ [0, 1]. The clusters contain consecutive exceedances of the time series over a threshold u separated by return intervals with consecutive non-exceedances. We derive geometric forms of asymptotically equal distributions of the normalized cluster and inter-cluster sizes that depend on θ. The inter-cluster size determines the number T 1(u) of inter-arrival times between observations of the process R t arising between two consecutive clusters. The cluster size is equal to the number T 2(u) of inter-arrival times within clusters. The inferences are valid when u is taken as a sufficiently high quantile of the process {R n }. The derived geometric models allow us to obtain the asymptotically equal mean of T 2(u) and other indices of clusters. Applications in telecommunication networks are discussed.
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Notes
- 1.
D(u n ) is satisfied if for any \(A \in \mathcal{I}_{1,l}(u_{n})\) and \(B \in \mathcal{I}_{l+s,n}(u_{n})\), where \(\mathcal{I}_{j,l}(u_{n})\) is the set of all intersections of the events of the form {R i ≤ u n } for j ≤ i ≤ l, and for some positive integer sequence {s n } such that s n = o(n), \(\vert P\{(A\bigcap B)\} - P\{A\}P\{B\}\vert \leq \alpha (n,s)\) holds and α(n, s n ) → 0 as n → ∞.
- 2.
The D ′ ′(u n )-condition [3, 6] states that if the stationary sequence {R t } satisfies the D(u n )-condition with \(u_{n} = a_{n}x + b_{n}\) and normalizing sequences a n > 0 and b n ∈ R such that for all x there exists μ ∈ R, σ > 0 and ξ ∈ R, such that
$$\displaystyle{n\left (1 - F(a_{n}x + b_{n})\right ) \rightarrow \left (1 + \frac{\xi (x-\mu )} {\sigma } \right )_{+}^{-1/\xi },\qquad \text{as}\qquad n \rightarrow \infty,}$$holds, where \((x)_{+} =\max (x, 0)\), then \(\lim _{n\rightarrow \infty }\sum _{j=2}^{r_{n}}P\{R_{j} \leq u_{n} < R_{j+1}\vert R_{1} > u_{n}\} = 0\), where r n = o(n), s n = o(n), α(n, s n ) → 0, (n∕r n )α(n, s n ) → 0 and s n ∕r n → 0 as n → ∞.
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Acknowledgements
This work was supported in part by the Russian Foundation for Basic Research, grant 13-08-00744 A.
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Markovich, N. (2014). Distributions of Clusters of Exceedances and Their Applications in Telecommunication Networks. In: Akritas, M., Lahiri, S., Politis, D. (eds) Topics in Nonparametric Statistics. Springer Proceedings in Mathematics & Statistics, vol 74. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0569-0_15
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