Volume 17, Number 4
Consider a finite renewal process in the sense that interrenewal times are positive i.i.d. variables and the total number of renewals is a random variable, independent of interrenewal times. A finite point process can be obtained by probabilistic sampling of the finite renewal process, where each renewal is sampled with a fixed probability and independently of other renewals. The problem addressed in this work concerns statistical inference of the original distributions of the total number of renewals and interrenewal times from a sample of i.i.d. finite point processes obtained by sampling finite renewal processes. This problem is motivated by traffic measurements in the Internet in order to characterize flows of packets (which can be seen as finite renewal processes) and where the use of packet sampling is becoming prevalent due to increasing link speeds and limited storage and processing capacities.
 Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. New York: Wiley.
 Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987). Regular variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge: Cambridge Univ. Press.
Mathematical Reviews (MathSciNet): MR898871
 Bøgsted, M. and Pitts, S.M. (2010). Decompounding random sums: A nonparametric approach. Ann. Inst. Statist. Math. 62 855–872.
 Buchmann, B. (2001). Decompounding: An estimation problem for the compound Poisson distribution. Ph.D. thesis, Univ. Hannover.
 Buchmann, B. and Grübel, R. (2003). Decompounding: An estimation problem for Poisson random sums. Ann. Statist. 31 1054–1074.
 Chabchoub, Y., Fricker, C., Guillemin, F. and Robert, P. (2010). On the statistical characterization of flows in Internet traffic with application to sampling. Comput. Commun. 31 103–112.
 Clegg, R.G., Landa, R., Haddadi, H., Rio, M. and Moore, A.W. (2008). Techniques for flow inversion on sampled data. In 2008 IEEE INFOCOM Workshops 1–6.
 Cox, D.R. and Isham, V. (1980). Point Processes. London: Chapman and Hall.
Mathematical Reviews (MathSciNet): MR598033
 Daley, D.J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes. Vol. I, 2nd ed. New York: Springer.
 Duffield, N., Lund, C. and Thorup, M. (2002). Properties and prediction of flow statistics from sampled packet streams. In Proc. ACM SIGCOMM Internet Measurement Workshop 159–171.
 Duffield, N., Lund, C. and Thorup, M. (2005). Estimating flow distributions from sampled flow statistics. IEEE/ACM Transactions on Networking 13 933–946.
 Estan, C. and Varghese, G. (2002). New directions in traffic measurement and accounting. In SIGCOMM’02. Conference on Applications, Technologies, Architectures, and Protocols for Computer Communications 323–336.
 Grübel, R. and Pitts, S.M. (1993). Nonparametric estimation in renewal theory. I. The empirical renewal function. Ann. Statist. 21 1431–1451.
 Hall, P. and Park, J. (2004). Nonparametric inference about service time distribution from indirect measurements. J. R. Stat. Soc. Ser. B. Stat. Methodol. 66 861–875.
 Hansen, M.B. and Pitts, S.M. (2006). Nonparametric inference from the M/G/1 workload. Bernoulli 12 737–759.
 Henrici, P. (1974). Applied and Computational Complex Analysis. New York: Wiley.
Mathematical Reviews (MathSciNet): MR372162
 Hohn, N. and Veitch, D. (2006). Inverting sampled traffic. IEEE/ACM Transactions on Networking 14 68–80.
 Hohn, N., Veitch, D. and Abry, P. (2003). Cluster processes: A natural language for network traffic. IEEE Trans. Signal Process. 51 2229–2244.
 Karr, A.F. (1991). Point Processes and Their Statistical Inference, 2nd ed. Probability: Pure and Applied 7. New York: Marcel Dekker.
 Pollard, D. (1984). Convergence of Stochastic Processes. New York: Springer.
Mathematical Reviews (MathSciNet): MR762984
 Robert, C.Y. and Segers, J. (2008). Tails of random sums of a heavy-tailed number of light-tailed terms. Insurance Math. Econom. 43 85–92.
 Tune, P. and Veitch, D. (2008). Towards optimal sampling for flow size estimation. In ACM SIGCOMM Internet Measurement Conference 243–256.
 Vakhania, N.N., Tarieladze, V.I. and Chobanyan, S.A. (1987). Probability Distributions on Banach Spaces. Mathematics and Its Applications (Soviet Series) 14. Dordrecht: D. Reidel Publishing.
 Yang, L. and Michailidis, G. (2007). Sampled based estimation of network traffic flow characteristics. In INFOCOM 2007. 26th IEEE International Conference on Computer Communications 1775–1783.