Weitere Kapitel dieses Buchs durch Wischen aufrufen
This chapter provides the necessary background to support the rest of the book. No attempt has been made to make this book really self-contained. The book will survey many recent results in the literature. We often include preliminary tools from publications. These preliminary tools may be still too difficult for many of the audience. Roughly, our prerequisite is the graduate-level course on random variables and processes.
Bitte loggen Sie sich ein, um Zugang zu diesem Inhalt zu erhalten
Sie möchten Zugang zu diesem Inhalt erhalten? Dann informieren Sie sich jetzt über unsere Produkte:
R. Qiu, Z. Hu, H. Li, and M. Wicks, Cognitiv Communications and Networking: Theory and Practice. John Wiley and Sons, 2012.
G. Lugosi, “Concentration-of-measure inequalities,” 2009.
A. Leon-Garcia, Probability, Statistics, and Random Processing for Electrical Engineering. Pearson-Prentice Hall, third edition ed., 2008.
T. Tao, Topics in Random Matrix Thoery. American Mathematical Society, 2012.
N. Nguyen, P. Drineas, and T. Tran, “Tensor sparsification via a bound on the spectral norm of random tensors,” arXiv preprint arXiv:1005.4732, 2010.
W. Hoeffding, “Probability inequalities for sums of bounded random variables,” Journal of the American Statistical Association, pp. 13–30, 1963.
A. Van Der Vaart and J. Wellner, Weak Convergence and Empirical Processes. Springer-Verlag, 1996.
F. Lin, R. Qiu, Z. Hu, S. Hou, J. Browning, and M. Wicks, “Generalized fmd detection for spectrum sensing under low signal-to-noise ratio,” IEEE Communications Letters, to appear.
E. Carlen, “Trace inequalities and quantum entropy: an introductory course,” Entropy and the quantum: Arizona School of Analysis with Applications, March 16–20, 2009, University of Arizona, vol. 529, 2010.
F. Zhang, Matrix Theory. Springer Ver, 1999.
K. Abadir and J. Magnus, Matrix Algebra. Cambridge Press, 2005.
D. S. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas. Princeton University Press, 2009.
J. A. Tropp, “User-friendly tail bounds for sums of random matrices.” Preprint, 2011.
N. J. Higham, Functions of Matrices: Theory and Computation. Society for Industrial and Applied Mathematics, 2008.
L. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, 2005.
R. Bhatia, Positive Definite Matrices. Princeton University Press, 2007.
R. Bhatia, Matrix analysis. Springer, 1997.
A. W. Marshall, I. Olkin, and B. C. Arnold, Inequalities: Theory of Majorization and Its Applications. Springer Verl, 2011.
D. Watkins, Fundamentals of Matrix Computations. Wiley, third ed., 2010.
M. Ledoux and M. Talagrand, Probability in Banach spaces. Springer, 1991.
J. Nelson, “Johnson-lindenstrauss notes,” tech. rep., Technical report, MIT-CSAIL, Available at http://web.mit.edu/minilek/www/jl_notes.pdf, 2010.
H. Rauhut, “Compressive sensing and structured random matrices,” Theoretical foundations and numerical methods for sparse recovery, vol. 9, pp. 1–92, 2010. MathSciNet
F. Krahmer, S. Mendelson, and H. Rauhut, “Suprema of chaos processes and the restricted isometry property,” arXiv preprint arXiv:1207.0235, 2012.
W. Bednorz and R. Latala, “On the suprema of bernoulli processes,” Comptes Rendus Mathematique, 2013.
B. Gnedenko and A. Kolmogorov, Limit Distributions for Sums Independent Random Variables. Addison-Wesley, 1954.
R. Vershynin, “A note on sums of independent random matrices after ahlswede-winter.” http://www-personal.umich.edu/~romanv/teaching/reading-group/ahlswede-winter.pdf. Seminar Notes.
T. Fine, Probability and Probabilistic Reasoning for Electrical Engineering. Pearson-Prentice Hall, 2006.
R. Oliveira, “Sums of random hermitian matrices and an inequality by rudelson,” Elect. Comm. Probab, vol. 15, pp. 203–212, 2010. MATH
T. Rockafellar, Conjugative duality and optimization. Philadephia: SIAM, 1974. CrossRef
D. Petz, “A suvery of trace inequalities.” Functional Analysis and Operator Theory, 287–298, Banach Center Publications, 30 (Warszawa), 1994. http://www.renyi.hu/~petz/pdf/64.pdf.
R. Vershynin, “Golden-thompson inequality.” http://www-personal.umich.edu/~romanv/teaching/reading-group/golden-thompson.pdf. Seminar Notes.
J. Tropp, “From joint convexity of quantum relative entropy to a concavity theorem of lieb,” in Proc. Amer. Math. Soc, vol. 140, pp. 1757–1760, 2012.
E. G. Effros, “A matrix convexity approach to some celebrated quantum inequalities,” vol. 106, pp. 1006–1008, National Acad Sciences, 2009.
T. Rockafellar, Conjugate duality and optimization. SIAM, 1974. Regional conference series in applied mathematics.
S. Boyd and L. Vandenberghe, Convex optimization. Cambridge Univ Pr, 2004.
V. I. Paulsen, Completely Bounded Maps and Operator Algebras. Cambridge Press, 2002.
T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: John Wiley.
P. Halmos, Finite-Dimensional Vector Spaces. Springer, 1958.
V. De la Peña and E. Giné, Decoupling: from dependence to independence. Springer Verlag, 1999.
R. Vershynin, “A simple decoupling inequality in probability theory,” May 2011.
P. Billingsley, Probability and Measure. Wiley, 2008.
R. Dudley, Real analysis and probability, vol. 74. Cambridge University Press, 2002.
D. L. Hanson and F. T. Wright, “A bound on tail probabilities for quadratic forms in independent random variables,” The Annals of Mathematical Statistics, pp. 1079–1083, 1971.
T. Tao, Topics in Random Matrix Theory. Amer Mathematical Society, 2012.
E. Wigner, “Distribution laws for the roots of a random hermitian matrix,” Statistical Theories of Spectra: Fluctuations, pp. 446–461, 1965.
M. Mehta, Random matrices, vol. 142. Academic press, 2004.
J. Wishart, “The generalised product moment distribution in samples from a normal multivariate population,” Biometrika, vol. 20, no. 1/2, pp. 32–52, 1928.
P. Hsu, “On the distribution of roots of certain determinantal equations,” Annals of Human Genetics, vol. 9, no. 3, pp. 250–258, 1939. CrossRef
A. Erdelyi, W. Magnus, Oberhettinger, and F. Tricomi, eds., Higher Transcendental Functions, Vol. 1–3. McGraw-Hill, 1953.
R. Vershynin, “Introduction to the non-asymptotic analysis of random matrices,” Arxiv preprint arXiv:1011.3027v5, July 2011.
D. Garling, Inequalities: a journey into linear analysis. Cambridge University Press, 2007.
V. Buldygin and S. Solntsev, Asymptotic behaviour of linearly transformed sums of random variables. Kluwer, 1997.
J. Kahane, Some random series of functions. Cambridge Univ Press, 2nd ed., 1985.
M. Rudelson, “Lecture notes on non-asymptotic theory of random matrices,” arXiv preprint arXiv:1301.2382, 2013.
V. Yurinsky, Sums and Gaussian vectors. Springer-Verlag, 1995.
U. Haagerup, “The best constants in the khintchine inequality,” Studia Math., vol. 70, pp. 231–283, 1981.
M. Talagrand, The generic chaining: upper and lower bounds of stochastic processes. Springer Verlag, 2005.
M. Talagrand, Upper and Lower Bounds for Stochastic Processes, Modern Methods and Classical Problems. Springer-Verlag, in press. Ergebnisse der Mathematik.
X. Fernique, “Régularité des trajectoires des fonctions aléatoires gaussiennes,” Ecole d’Eté de Probabilités de Saint-Flour IV-1974, pp. 1–96, 1975.
R. Bhattacharya and R. Rao, Normal approximation and asymptotic expansions, vol. 64. Society for Industrial & Applied, 1986.
L. Chen, L. Goldstein, and Q. Shao, Normal Approximation by Stein’s Method. Springer, 2010.
A. Kirsch, An introduction to the mathematical theory of inverse problems, vol. 120. Springer Science+ Business Media, 2011.
D. Porter and D. S. Stirling, Integral equations: a practical treatment, from spectral theory to applications, vol. 5. Cambridge University Press, 1990.
A. Soshnikov, “Level spacings distribution for large random matrices: Gaussian fluctuations,” Annals of mathematics, pp. 573–617, 1998.
- Mathematical Foundation
- Springer New York
- Chapter 1
Neuer Inhalt/© Filograph | Getty Images | iStock