Tauberian theorems for matrix regular variation
Authors:
M. M. Meerschaert and H.P. Scheffler
Journal:
Trans. Amer. Math. Soc. 365 (2013), 22072221
MSC (2010):
Primary 40E05; Secondary 44A10, 26A12
Published electronically:
October 3, 2012
MathSciNet review:
3009656
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Karamata's Tauberian theorem relates the asymptotics of a nondecreasing rightcontinuous function to that of its LaplaceStieltjes transform, using regular variation. This paper establishes the analogous Tauberian theorem for matrixvalued functions. Some applications to time series analysis are indicated.
 1.
A.A. Balkema (1973) Monotone Transformations and Limit Laws. Mathematical Centre Tracts 45, Mathematisch Centrum, Amsterdam, 1973. MR 0334307 (48:12626)
 2.
Guus
Balkema and Paul
Embrechts, High risk scenarios and extremes, Zurich Lectures
in Advanced Mathematics, European Mathematical Society (EMS), Zürich,
2007. A geometric approach. MR 2372552
(2009d:60002)
 3.
Ph. Barbe and W.P. McCormick (2011) Invariance principles for some FARIMA and nonstationary linear processes in the domain of a stable distribution. Probab. Theory Relat. Fields, to appear, DOI 10.1007/s0044001103743.
 4.
Ph.
Barbe and W.
P. McCormick, Veraverbeke’s theorem at large: on the maximum
of some processes with negative drift and heavy tail innovations,
Extremes 14 (2011), no. 1, 63–125. MR 2775871
(2012g:60149), 10.1007/s1068701001039
 5.
N.
H. Bingham, C.
M. Goldie, and J.
L. Teugels, Regular variation, Encyclopedia of Mathematics and
its Applications, vol. 27, Cambridge University Press, Cambridge,
1989. MR
1015093 (90i:26003)
 6.
Peter
J. Brockwell and Richard
A. Davis, Time series: theory and methods, 2nd ed., Springer
Series in Statistics, SpringerVerlag, New York, 1991. MR 1093459
(92d:62001)
 7.
Charles
W. Curtis, Linear algebra: an introductory approach, 3rd ed.,
Allyn and Bacon, Inc., Boston, Mass., 1974. MR 0345978
(49 #10704)
 8.
William
Feller, An introduction to probability theory and its applications.
Vol. I, Third edition, John Wiley & Sons, Inc., New
YorkLondonSydney, 1968. MR 0228020
(37 #3604)
 9.
L.
Jódar and J.
C. Cortés, Some properties of gamma and beta matrix
functions, Appl. Math. Lett. 11 (1998), no. 1,
89–93. MR
1490386 (99b:33002), 10.1016/S08939659(97)001390
 10.
Mark
M. Meerschaert, Regular variation in
𝑅^{𝑘}, Proc. Amer. Math.
Soc. 102 (1988), no. 2, 341–348. MR 920997
(89b:26014), 10.1090/S00029939198809209975
 11.
Mark
M. Meerschaert and HansPeter
Scheffler, Limit distributions for sums of independent random
vectors, Wiley Series in Probability and Statistics: Probability and
Statistics, John Wiley & Sons, Inc., New York, 2001. Heavy tails in
theory and practice. MR 1840531
(2002i:60047)
 1.
 A.A. Balkema (1973) Monotone Transformations and Limit Laws. Mathematical Centre Tracts 45, Mathematisch Centrum, Amsterdam, 1973. MR 0334307 (48:12626)
 2.
 A.A. Balkema and P. Embrechts (2007) High Risk Scenarios and Extremes. A geometric approach. Zurich Lecture Notes. MR 2372552 (2009d:60002)
 3.
 Ph. Barbe and W.P. McCormick (2011) Invariance principles for some FARIMA and nonstationary linear processes in the domain of a stable distribution. Probab. Theory Relat. Fields, to appear, DOI 10.1007/s0044001103743.
 4.
 Ph. Barbe and W.P. McCormick (2011) Veraverbeke's theorem at large: on the maximum of some processes with negative drift and heavy tail innovations. Extremes, 14(1), 63125. MR 2775871
 5.
 N.H. Bingham, C.M. Goldie and J.L. Teugels (1989) Regular Variation. 2nd ed., Cambridge University Press. MR 1015093 (90i:26003)
 6.
 P. Brockwell and R. Davis (1991) Time Series: Theory and Methods, 2nd Ed., SpringerVerlag, New York. MR 1093459 (92d:62001)
 7.
 C. Curtis (1974) Linear Algebra. 3rd Ed., Allyn and Bacon, Boston. MR 0345978 (49:10704)
 8.
 W. Feller (1968) An Introduction to Probability Theory and Its Applications Vol. 2. Wiley Interscience, New York. MR 0228020 (37:3604)
 9.
 L. Jódar and J.C. Cortés (1998) Some properties of gamma and beta matrix functions. Appl. Math. Lett. 11, 8993. MR 1490386 (99b:33002)
 10.
 M.M. Meerschaert (1988) Regular Variation in . Proc. Amer. Math. Soc. 102, 341348. MR 920997 (89b:26014)
 11.
 M.M. Meerschaert and H.P. Scheffler (2001)
Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice. Wiley Interscience, New York. MR 1840531 (2002i:60047)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2010):
40E05,
44A10,
26A12
Retrieve articles in all journals
with MSC (2010):
40E05,
44A10,
26A12
Additional Information
M. M. Meerschaert
Affiliation:
Department of Statistics and Probability, Michigan State University, East Lansing, Michigan 48824
Email:
mcubed@stt.msu.edu
H.P. Scheffler
Affiliation:
Fachbereich Mathematik, Universität Siegen, 57068 Siegen, Germany
Email:
scheffler@mathematik.unisiegen.de
DOI:
http://dx.doi.org/10.1090/S000299472012057515
Received by editor(s):
September 12, 2011
Published electronically:
October 3, 2012
Additional Notes:
Research of the first author was partially supported by NSF grants DMS1025486, DMS0803360, and NIH grant R01EB012079.
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
