##
**Strong approximations in probability and statistics.**
*(English)*
Zbl 0539.60029

Probability and Mathematical Statistics. New York-San Francisco-London: Academic Press, a Subsidiary of Harcourt Brace Jovanovich, Publishers; Budapest: Akadémiai Kiadó. 284 p. $ 35.50 (1981).

In recent years there has been an increasing interest in so-called strong approximations. Look at the following false statement: Convergence in distribution implies a.s.-convergence. This statement can be made into a correct one if it is modified in the following way. If \(X_ n\) is a sequence of random elements with values in a suitable space and which converges in distribution to X, then one can construct a probability space and variables \(X_ n\!'\) which have the same (individual) laws as \(X_ n\) and which converge almost surely.

The present book centers on such constructions mainly in connection with Donsker’s invariance principle. Strong approximations with rates had been obtained by Strassen with the Skorohod imbedding which represents sums of i.i.d. random variables by means of stopped Brownian motion. Later on, it became known that this method cannot give the optimal approximation rates for sums of random variables with higher moments. The famous construction of J. Komlós, P. Major and G. Tusnády which was published in Z. Wahrscheinlichkeitstheor. Verw. Geb. 32, 111-131 (1975; Zbl 0308.60029) and ibid. 34, 33-58 (1976; Zbl 0307.60045) gave the optimal rates. The book gives a very profound introduction into these topics including some very recent contributions. It starts with an introduction into Wiener processes including two parameters processes and a detailed discussion of the fluctuation behavior of such processes which has been obtained by the authors in recent years.

The second chapter is an introduction to the strong approximation techniques starting with the Skorohod constructions and giving the main ideas of the very complicated Komlós-Major-Tusnády method. This is then applied in chapter 3 to a detailed study of partial sum processes and in chapters 4 and 5 to empirical and quantile processes. The last chapters discuss some additional topics including random limit theorems (i.e. limit theorems for randomly summed random variables).

The book contains many topics which did not appear before in book form. It achieves the difficult task to be an excellent introduction into this field for the beginner and also a source of material and methods for the specialists.

The present book centers on such constructions mainly in connection with Donsker’s invariance principle. Strong approximations with rates had been obtained by Strassen with the Skorohod imbedding which represents sums of i.i.d. random variables by means of stopped Brownian motion. Later on, it became known that this method cannot give the optimal approximation rates for sums of random variables with higher moments. The famous construction of J. Komlós, P. Major and G. Tusnády which was published in Z. Wahrscheinlichkeitstheor. Verw. Geb. 32, 111-131 (1975; Zbl 0308.60029) and ibid. 34, 33-58 (1976; Zbl 0307.60045) gave the optimal rates. The book gives a very profound introduction into these topics including some very recent contributions. It starts with an introduction into Wiener processes including two parameters processes and a detailed discussion of the fluctuation behavior of such processes which has been obtained by the authors in recent years.

The second chapter is an introduction to the strong approximation techniques starting with the Skorohod constructions and giving the main ideas of the very complicated Komlós-Major-Tusnády method. This is then applied in chapter 3 to a detailed study of partial sum processes and in chapters 4 and 5 to empirical and quantile processes. The last chapters discuss some additional topics including random limit theorems (i.e. limit theorems for randomly summed random variables).

The book contains many topics which did not appear before in book form. It achieves the difficult task to be an excellent introduction into this field for the beginner and also a source of material and methods for the specialists.

Reviewer: E.Bolthausen

### MSC:

60F15 | Strong limit theorems |

60F05 | Central limit and other weak theorems |

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