Heavy-traffic approximations for fractionally integrated random walks in the domain of attraction of a non-Gaussian stable distribution. (English) Zbl 1254.60035
Let \((X_i)_{i\geq 1}\) be a sequence of i.i.d. mean zero random variables having a distribution function \(F\). Set \(S_0=0\) and \(S_n=\sum_{0\leq i <n}g_iX_{n-i}\), \(n\in\mathbb{N}\), where the series \(g(x)=\sum_{i\geq 0}g_ix^i\) has a radius of convergence of at least \(1\). This \((g,F)\)-process \((S_n)_{n\geq 0}\) comprises, e.g., an autoregressive moving average (ARMA) one. Under specified conditions (among them \(g_n =cn^{\gamma -1}(1+o(n^{-\varepsilon}))\) as \(n\to \infty\) for some \(c,\varepsilon >0\) and \(\gamma >1\); \(F\) is in the domain of attraction of a stable law of index \(\alpha\)), one can prove that appropriately normalized \(\sup_{n\geq 1}(S_n - a\sum_{0\leq i<n}g_i)\) converges weakly as \(a\to 0\) to a certain random variable defined by means of a fractional Lévy stable process.
Reviewer: Alexander V. Bulinski (Moskva)
MSC:
| 60F17 | Functional limit theorems; invariance principles |
| 60G52 | Stable stochastic processes |
| 60G22 | Fractional processes, including fractional Brownian motion |
| 60K25 | Queueing theory (aspects of probability theory) |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 60G70 | Extreme value theory; extremal stochastic processes |
| 62P20 | Applications of statistics to economics |
Keywords:
heavy traffic; fractionally integrated autoregressive-moving average (FARIMA) process; fractional Lévy stable processReferences:
| [1] | Ph. Barbe, W.P. McCormick, Invariance principle for some FARIMA and nonstationary linear processes in the domain of a stable law, Probab. Theory Related Fields (2010) (in press). arXiv:1007.0576; Ph. Barbe, W.P. McCormick, Invariance principle for some FARIMA and nonstationary linear processes in the domain of a stable law, Probab. Theory Related Fields (2010) (in press). arXiv:1007.0576 · Zbl 1271.60050 |
| [2] | Bassan, B.; Bona, E., Moments of stochastic processes governed by Poisson random measures, Comment. Math. Univ. Carolin., 31, 337-343 (1990) · Zbl 0724.60051 |
| [3] | Bingham, N. H.; Goldie, C. M.; Teugels, J. L., Regular Variation (1989), Cambridge University Press · Zbl 0667.26003 |
| [4] | Braverman, M.; Mikosch, T.; Samorodnitsky, G., Tail probabilities of subadditive functionals of Lévy processes, Ann. Appl. Probab., 12, 69-100 (2002) · Zbl 1035.60045 |
| [5] | Csörgő, M.; Csörgő, S.; Horvàth, L.; Mason, D., Normal and stable convergence of integral functions of the empirical distribution function, Ann. Probab., 14, 86-118 (1986) · Zbl 0589.60030 |
| [6] | Dieker, A. B., Conditional limit theorems for queues with Gaussian input, a weak convergence approach, Stoch. Proc. Appl., 115, 849-873 (2005) · Zbl 1073.60085 |
| [7] | Ganesh, A.; O’Connell, N., A large deviation principle with queueing applications, Stoch. Stoch. Rep., 73, 250036 (2002) · Zbl 1006.60021 |
| [8] | Ganesh, A.; O’Connell, N.; Wischik, D., Big Queues (2004), Springer · Zbl 1044.60001 |
| [9] | Geffroy, J., Contribution à la théorie des valeurs extrêmes, Publ. Inst. Statist. Univ. Paris, 7-8, 37-185 (1958-1959) |
| [10] | Hall, P.; Weissman, I., On the estimation of extreme tail probabilities, Ann. Statist., 25, 1311-1326 (1997) · Zbl 0880.62036 |
| [11] | Kiefer, J., Iterated logarithm analogues for sample quantiles when \(p_n \downarrow 0\), (Proc. Sixth Berkeley Sympos. on Math. Statist. and Probab., vol. 1 (1972)), 227-244 · Zbl 0264.62015 |
| [12] | Kingman, J. F.C., The single server queue in heavy traffic, Proc. Cambridge Philos. Soc., 57, 902-904 (1961) · Zbl 0114.11703 |
| [13] | Kingman, J. F.C., On queues in heavy traffic, J. R. Stat. Soc. Ser. B, 24, 383-392 (1962) · Zbl 0127.10003 |
| [14] | Kingman, J. F.C., The heavy traffic approximation in the theory of queues, (Smith, W. L.; Wilkinson, W. E., Proc. Symp. on Congestion Theory (1965), University of North Carolina Press), 137-159 · Zbl 0189.51604 |
| [15] | Privault, N., Moment identities for Poisson-Skorokhod integrals and application to measure invariance, C. R. Acad. Sci., Paris, Sér. I, 347, 1071-1074 (2009) · Zbl 1179.60035 |
| [16] | N. Privault, Invariance of Poisson measures under random transformations, 2010. arXiv:1004.2588v1; N. Privault, Invariance of Poisson measures under random transformations, 2010. arXiv:1004.2588v1 · Zbl 1278.60084 |
| [17] | Prohorov, Yu. V., Transition phenomena in queueing processes, I, Litov. Fiz. Sb., 3, 199-205 (1963), (in Russian) |
| [18] | Protter, Ph., Stochastic Integration and Differential Equations, A New Approach (1992), Springer |
| [19] | Pruitt, W., The growth of random walks and Lévy processes, Ann. Probab., 9, 948-956 (1981) · Zbl 0477.60033 |
| [20] | Resnick, S. I., Heavy-Tail Phenomena, Probabilistic and Statistical Modeling (2007), Springer · Zbl 1152.62029 |
| [21] | S. Shneer, V. Wachtel, Heavy-traffic analysis of the maximum of an asymptotically stable random walk, 2009. arXiv:0902.2185; S. Shneer, V. Wachtel, Heavy-traffic analysis of the maximum of an asymptotically stable random walk, 2009. arXiv:0902.2185 · Zbl 1229.60059 |
| [22] | Shorack, G. R.; Wellner, J. A., Linear bounds on the empirical distribution function, Ann. Probab., 6, 349-353 (1978) · Zbl 0376.60034 |
| [23] | Shorack, G. R.; Wellner, J. A., Empirical Processes with Applications to Statistics (1986), Wiley · Zbl 1170.62365 |
| [24] | Szczotka, W.; Woykzyński, W. A., Distribution of suprema of Lévy processes via the heavy traffic invariance principle, Probab. Math. Statist., 23, 251-272 (2003) · Zbl 1048.60038 |
| [25] | Whitt, W., Stochastic-Process Limits, An Introduction to Stochastic-Process Limits and their Applications to Queues (2002), Springer · Zbl 0993.60001 |
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