## Large deviations of inverse processes with nonlinear scalings.(English)Zbl 0939.60011

For a right-continuous, nonnegative, nondecreasing stochastic process $$(Z_t)_{t\geq 0}$$ with $$\lim_{t\to\infty }Z_t=\infty$$ we define the inverse process $$T_z=\inf\{t\geq 0, Z_t\geq z\}$$, $$z\geq 0$$, which is a left-continuous, nonnegative, nondecreasing process with $$T_0=0$$ and $$\lim_{t\to\infty }T_t=\infty$$. Then, under regularity conditions, $$u(Z_t)/w(t)$$ satisfies the large deviation principle (LDP) with a scaling function $$v(t)$$ and a rate function $$I$$ if and only if $$w(T_z)/u(z)$$ satisfies the LDP with the scaling function $$v\circ w^{-1}\circ u(z)$$ and the rate function $$J(x)=f(x)I(1/x)$$ where $$f(x)=x^{i(f)}$$ with $$i(f)$$ being the index of the regularly varying function $$v\circ w^{-1}$$. An application to queueing models as well as examples concerning fractional Gaussian noise, time-transformed stationary processes, compound processes and renewal theory are also given.

### MSC:

 60F10 Large deviations 60K25 Queueing theory (aspects of probability theory) 60G18 Self-similar stochastic processes
Full Text:

### References:

 [1] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Ency clopedia of Mathematics and Its Applications 27. Cambridge Univ. Press. · Zbl 0617.26001 [2] Borovkov, A. A. (1976). Stochastic Processes in Queueing Theory. Springer, New York. · Zbl 0319.60057 [3] Chang, C.-S. (1994). Stability, queue length and delay of deterministic and stochastic queueing networks. IEEE Trans. Automat. Control 39 913-931. · Zbl 0818.90050 [4] Crosby, S., Leslie, I., Lewis, J. T., O’Connell, N., Russell, R. and Toomey, F. (1995). By passing modeling: an investigation of entropy as a traffic descriptor in the Fairisle ATM network. In Proceedings of the 12th IEE UK Teletraffic Sy mposium, Old Windsor, 15-17 March 1995. [5] Dembo, A. and Zeitouni, O. (1993). Large Deviation Techniques and Applications. Jones and Bartlett, Boston. · Zbl 0793.60030 [6] Duffield, N. G. (1994). Exponential bounds for queues with Markovian arrivals. Queueing Sy stems 17 413-430. · Zbl 0811.60085 [7] Duffield, N. G. (1996). Economies of scale in queues with sources having power-law large deviation scalings. J. Appl. Probab. 33 840-857. JSTOR: · Zbl 0866.60078 [8] Duffield, N. G. (1996). On the relevance of long-tailed durations for the statistical multiplexing of large aggregations. In Proceedings of the 34th Annual Allerton Conference on Communication, Control, and Computing (S. P. Mey n and W. K. Jenkins, eds.) 741-750. Univ. Illinois, Urbana-Champaign. [9] Duffield, N. G. and O’Connell, N. (1995). Large deviations and overflow probabilities for the general single-server queue, with applications. Math. Proc. Cambridge. Philos. Soc. 118 363-374. · Zbl 0840.60087 [10] Duffield, N. G., Lewis, J. T., O’Connell, N., Russell, R. and Toomey, F. (1995). Entropy of ATM traffic streams: a tool for estimating QoS parameters. IEEE J. Selected Areas in Comm. 13 981-990. [11] Gly nn, P. W. and Whitt, W. (1994). Large deviations behavior of counting processes and their inverses. Queueing Sy stems 17 107-128. · Zbl 0805.60023 [12] Gly nn, P. W. and Whitt, W. (1994). Logarithmic asy mptotics for steady-state tail probabilities in a single-server queue. J. Appl. Probab. 31A 131-159. JSTOR: · Zbl 0805.60093 [13] Hobson, E. W. (1927). The Theory of Functions of a Real Variable and the Theory of Fourier’s series 1. Cambridge Univ. Press. · JFM 53.0226.01 [14] Kesidis, G., Walrand, J. and Chang, C. S. (1993). Effective bandwidths for multiclass Markov fluids and other ATM Sources. IEEE/ACM Trans. Networking 1 424-428. [15] Leland, W. E., Taqqu, M. S., Willinger, W. and Wilson, D. V. (1993). On the self-similar nature of Ethernet traffic. ACM SIGCOMM Computer Communications Review 23 183-193. [16] Mandelbrot, B. B. and Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 422-437. JSTOR: · Zbl 0179.47801 [17] Massey, W. A. and Whitt, W. (1994). Unstable asy mptotics for nonstationary queues. Math. Oper. Res. 19 267-291. JSTOR: · Zbl 0801.60087 [18] Norros, I. (1994). A storage model with self-similar input. Queueing Sy stems 16 387-396. · Zbl 0811.68059 [19] Parulekar, M. and Makowski, A. (1996). Tail probabilities for a multiplexer with selfsimilar traffic. In Proceedings of the IEEE INFOCOM’96, 1452-1459, IEEE, New York. [20] Puhalskii, A. and Whitt, W. (1997). Functional large deviations principles for first passage time processes. Ann. Appl. Probab. 7 362-381. · Zbl 0885.60023 [21] Rockafellar, R. T. (1970). Convex Analy sis. Princeton Univ. Press. · Zbl 0193.18401 [22] Russell, R. (1995). The large deviations of inversely related processes. Technical Report DIAS-APG 95-38, Dublin Inst. Advanced Studies.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.