Non-Gaussian semi-stable distributions and their statistical applications

RITWIK CHAUDHURI: NON-GAUSSIAN SEMI-STABLE DISTRIBUTIONS AND THEIR STATISTICAL APPLICATIONS (Under the direction of Vladas Pipiras) The dissertation is motivated by problems arising in modern communication networks such as the Internet. Over these networks, information is sent in the form of data packets which are further grouped into flows. For example, a flow can be associated with a certain (document, music, movie or other) file. Knowing the structure of flows is of great interest to network operators and networking researchers. One quantity of particular interest is the distribution of flow sizes (the number of packets in a flow). Each packet carries information on the flow it belongs to. Hence, examining all packets allows reconstructing and studying the associated flows. Examining all packets, however, is becoming cumbersome due to the ever increasing amount of data and processing costs. To overcome these issues, packet sampling has become prevalent. One common sampling scheme is probabilistic sampling wherein each packet is sampled independently and with the same probability. The basic problem then becomes inference of the characteristics of original flows (e.g. the flow size distribution) from sampled packets (forming sampled flows). This problem, known as an inversion problem, has attracted much attention in the networking community. In particular, a well-known nonparametric estimator of the flow size distribution is available under probabilistic sampling, based on sampled packets and sampled flows. From the application perspective, the focus of the dissertation is on some statistical properties of this nonparametric estimator. Under

[1]  D. Applebaum.  Stable non-Gaussian random processes , 1995, The Mathematical Gazette.

[2]  N. H. Bingham,et al.  Regular variation in more general settings , 1987 .

[3]  Victor M. Kruglov,et al.  On the Extension of the Class of Stable Distributions , 1973 .

[4]  Discrete semi-stable distributions , 2004 .

[5]  S. Csörgő,et al.  Merging asymptotic expansions for cooperative gamblers in generalized St. Petersburg games , 2008 .

[6]  佐藤 健一.  Lévy processes and infinitely divisible distributions , 2013 .

[7]  V. Zolotarev.  One-dimensional stable distributions , 1986 .

[8]  D. Mason,et al.  A probabilistic approach to the asymptotic distribution of sums of independent, identically distributed random variables , 1988 .

[9]  S. Csőrgő,et al.  Merging to semistable laws , 2003 .

[10]  R. Shimizu.  On the domain of partial attraction of semi-stable distributions , 1970 .

[11]  Mark M. Meerschaert,et al.  A simple robust estimation method for the thickness of heavy tails , 1998 .

[12]  S. Csörgö.  An approximation to infinitely divisible laws , 1995 .

[13]  Yu. S. Khokhlov,et al.  The Domains of Attraction of Semistable Laws , 1996 .

[14]  S. Csőrgő.  A probabilistic approach to domains of partial attraction , 1990 .

[15]  R. N. Pillai.  Semi Stable Laws as Limit Distributions , 1971 .

[16]  Nicolas Hohn,et al.  Inverting sampled traffic , 2003, IMC '03.

[17]  Ward Whitt,et al.  The Fourier-series method for inverting transforms of probability distributions , 1992, Queueing Syst. Theory Appl..

[18]  Knut K. Aase,et al.  On the St. Petersburg Paradox , 1998 .

[19]  Carsten Lund,et al.  Estimating flow distributions from sampled flow statistics , 2003, SIGCOMM '03.

[20]  Gordon Simons,et al.  A strong law of large numbers for trimmed sums, with applications to generalized St. Petersburg games☆ , 1996 .

[21]  M. Meerschaert,et al.  SEMISTABLE LÉVY MOTION , 2002 .

[22]  V. Pipiras,et al.  Probabilistic sampling of finite renewal processes , 2011, 1201.1076.

[23]  Kouji Yamamuro,et al.  Tail behaviors of semi-stable distributions , 2012 .

[24]  Lili Yang,et al.  Sampled Based Estimation of Network Traffic Flow Characteristics , 2007, IEEE INFOCOM 2007 - 26th IEEE International Conference on Computer Communications.