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Random Matrices and Non-Commutative Probability (Hardcover)
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Random Matrices and Non-Commutative Probability (Hardcover)
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This is an introductory book on Non-Commutative Probability or Free
Probability and Large Dimensional Random Matrices. Basic concepts
of free probability are introduced by analogy with classical
probability in a lucid and quick manner. It then develops the
results on the convergence of large dimensional random matrices,
with a special focus on the interesting connections to free
probability. The book assumes almost no prerequisite for the most
part. However, familiarity with the basic convergence concepts in
probability and a bit of mathematical maturity will be helpful.
Combinatorial properties of non-crossing partitions, including the
Moebius function play a central role in introducing free
probability. Free independence is defined via free cumulants in
analogy with the way classical independence can be defined via
classical cumulants. Free cumulants are introduced through the
Moebius function. Free product probability spaces are constructed
using free cumulants. Marginal and joint tracial convergence of
large dimensional random matrices such as the Wigner, elliptic,
sample covariance, cross-covariance, Toeplitz, Circulant and Hankel
are discussed. Convergence of the empirical spectral distribution
is discussed for symmetric matrices. Asymptotic freeness results
for random matrices, including some recent ones, are discussed in
detail. These clarify the structure of the limits for joint
convergence of random matrices. Asymptotic freeness of independent
sample covariance matrices is also demonstrated via embedding into
Wigner matrices. Exercises, at advanced undergraduate and graduate
level, are provided in each chapter.
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