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**Probability metrics and the stability of stochastic models.**
*(English)*
Zbl 0744.60004

Wiley Series in Probability and Mathematical Statistics. Applied Probability and Statistics. Chichester etc.: John Wiley & Sons Ltd.. xiv, 494 p. (1991).

Approximation and stability problems are basic problems in probability theory and its applications. The author makes an impressive and successful attempt to explain that it is useful to know a great variety of probability metrics and to know about their structural and topological properties in order to solve these problems. The book contains a wealth of recent or new theoretical results and applications, ranging from problems on the rate of convergence in limit theorems to the qualitative and quantitative analysis of the stability of stochastic models as e.g. the stability of queueing models or the stability of characterization results in probability theory. Further applications are to approximation problems in risk theory and operations research. It unifies many seemingly different problems in probability theory.

The treatment of relations between different probability metrics is based on dual and explicit solutions of the classical Monge-Kantorovich type problems. Some classical results in this direction are the Strassen- Dudley representation of the Prokhorov metric, the Kantorovich-Rubinstein theorem and the representation of the total variation metric due to Dobrushin, related to many coupling arguments. There are interesting recent developments on multivariate coupling problems, generalizations of Skorokhod’s a.s. representation theorem, on Glivenko-Cantelli type results on uniform convergence and on Fréchet-bounds.

Some basic notions and results in the theory of probability metrics were introduced by Zolotarev. The present book elaborates this approach in full depth and demonstrates its potential and actual usefulness. It enriches the probability theory by new results and new directions of work, poses some challenging open problems and establishes the theory of probability metrics as a valuable subject in probability theory. Without doubt it will be a basic orientation and reference point for future work in the field of approximation problems in probability theory. The book is well organized and very well readable. The author can be congratulated to his authentic and important work.

The treatment of relations between different probability metrics is based on dual and explicit solutions of the classical Monge-Kantorovich type problems. Some classical results in this direction are the Strassen- Dudley representation of the Prokhorov metric, the Kantorovich-Rubinstein theorem and the representation of the total variation metric due to Dobrushin, related to many coupling arguments. There are interesting recent developments on multivariate coupling problems, generalizations of Skorokhod’s a.s. representation theorem, on Glivenko-Cantelli type results on uniform convergence and on Fréchet-bounds.

Some basic notions and results in the theory of probability metrics were introduced by Zolotarev. The present book elaborates this approach in full depth and demonstrates its potential and actual usefulness. It enriches the probability theory by new results and new directions of work, poses some challenging open problems and establishes the theory of probability metrics as a valuable subject in probability theory. Without doubt it will be a basic orientation and reference point for future work in the field of approximation problems in probability theory. The book is well organized and very well readable. The author can be congratulated to his authentic and important work.

Reviewer: L.Rüschendorf (Münster)

### MSC:

60A10 | Probabilistic measure theory |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |