##
**Probabilities on the Heisenberg group: limit theorems and Brownian motion.**
*(English)*
Zbl 0870.60007

Lecture Notes in Mathematics. 1630. Berlin: Springer. viii, 139 p. (1996).

After certain aspects of probability theory had been unified within the setting of general locally compact groups the main engagement of researchers working in the field changed to deepening the available results for special classes of groups. It was not surprising that this new development started in France where Bourbaki’s spirit of utmost generality had somewhat faded in favor of a more problem-oriented trend of investigation. More precisely, in the succession of the monographs of U. Grenander [“Probabilities on algebraic structures” (1963; Zbl 0131.34804)], K. R. Parthasarathy [“Probability measures on metric spaces” (1967; Zbl 0153.19101)] and the reviewer [“Probability measures on locally compact groups” (1977; Zbl 0376.60002)] the structurally oriented probabilists were able to help themselves to rich collections of particular examples as they were contained in the Lecture Notes of Y. Guivarc’h, M. Keane and B. Roynette [“Marches aléatoires sur les groupes de Lie” (1977; Zbl 0367.60081)] and P. Diaconis [“Group representations in probability and statistics” (1988; Zbl 0695.60012)] on random walks on Lie groups and on group representations in probability and statistics, respectively.

Now, more than 20 years after the appearance of the reviewer’s monograph up-to-date treatments of parts of the subject are highly desirable. The book under review deals with probabilities on the Heisenberg group \(\mathbb{H}\), in the not too far future we shall have access to a book by W. Hazod and the late E. Siebert on stable convolution semigroups on groups and vector spaces. The present Lecture Notes have been completed in the course of the author’s “Habilitation” at the University of Bern. They are conceived expositorily but contain a large portion of original work accomplished by the author since his “Dissertation” in 1991. The author masters a great deal of literature on only 139 pages and at the same time manages to introduce the reader to a fascinating section of structural probability theory. The subtitle of the book which reads “Limit theorems and Brownian motion” refers to almost disjoint topics. Still the author succeeds at least methodically in relating properties of Brownian motion on \(\mathbb{H}\) (Chapter 2) to weak and almost sure limit theorems for more general semigroups, measures and random variables within the framework of \(\mathbb{H}\) (Chapter 3). In fact, Chapter 2 is mainly devoted to the work of L. Gallardo [in: Probability measures on groups, Lect. Notes Math. 928, 96-120 (1982; Zbl 0483.60072)] on the potential theory of Brownian motion (capacities, Wiener test, Poincaré’s criterion). But also the contributions of G. Pap (1992) and P. Ohring [Proc. Am. Math. Soc. 118, No. 4, 1313-1318 (1993; Zbl 0797.43003)] on the Lindeberg and Lyapunov theorems, and those of P. Crépel and B. Roynette [Z. Wahrscheinlichkeitstheorie Verw. Geb. 39, 217-229 (1977; Zbl 0342.60028)] on the iterated logarithm (for \(\mathbb{H}\)-valued random variables) receive special attention. In Chapter 3 the author touches even more upon his own work on the subject. He discusses the lightly trimmed products of \(\mathbb{H}\)-valued random variables with emphasis on the main theorem of his Ph. D. thesis of 1991, the Marcinkiewicz-Zygmund law of large numbers, non-classical laws of the iterated logarithm [mainly following the work of H.-P. Scheffler, Publ. Math. 47, No. 3/4, 377-391 (1995; Zbl 0861.60015) and Stat. Probab. Lett. 24, No. 3, 187-192 (1995; Zbl 0832.60011)], and the two-series theorem.

The book adds favorably to the literature on recent advances in probability theory on groups. It is clearly written with an elaborate introduction and a very useful first chapter on basic notions and technical preparations of the method. The rich bibliography of 176 items goes far beyond the references actually employed in the text. As for the choice of the material treated in the book, the author aimed at describing the recent state of the art by orienting himself at his own contributions. Fortunately, he did not try to update the pioneering monographs of the 60s; such an enterprise would have exceeded the habitual size of a “Habilitationsschrift” and would certainly have spoiled the attractivity immanent in this valuable handy volume that will doubtlessly encourage further research interest in probability theory on groups.

Now, more than 20 years after the appearance of the reviewer’s monograph up-to-date treatments of parts of the subject are highly desirable. The book under review deals with probabilities on the Heisenberg group \(\mathbb{H}\), in the not too far future we shall have access to a book by W. Hazod and the late E. Siebert on stable convolution semigroups on groups and vector spaces. The present Lecture Notes have been completed in the course of the author’s “Habilitation” at the University of Bern. They are conceived expositorily but contain a large portion of original work accomplished by the author since his “Dissertation” in 1991. The author masters a great deal of literature on only 139 pages and at the same time manages to introduce the reader to a fascinating section of structural probability theory. The subtitle of the book which reads “Limit theorems and Brownian motion” refers to almost disjoint topics. Still the author succeeds at least methodically in relating properties of Brownian motion on \(\mathbb{H}\) (Chapter 2) to weak and almost sure limit theorems for more general semigroups, measures and random variables within the framework of \(\mathbb{H}\) (Chapter 3). In fact, Chapter 2 is mainly devoted to the work of L. Gallardo [in: Probability measures on groups, Lect. Notes Math. 928, 96-120 (1982; Zbl 0483.60072)] on the potential theory of Brownian motion (capacities, Wiener test, Poincaré’s criterion). But also the contributions of G. Pap (1992) and P. Ohring [Proc. Am. Math. Soc. 118, No. 4, 1313-1318 (1993; Zbl 0797.43003)] on the Lindeberg and Lyapunov theorems, and those of P. Crépel and B. Roynette [Z. Wahrscheinlichkeitstheorie Verw. Geb. 39, 217-229 (1977; Zbl 0342.60028)] on the iterated logarithm (for \(\mathbb{H}\)-valued random variables) receive special attention. In Chapter 3 the author touches even more upon his own work on the subject. He discusses the lightly trimmed products of \(\mathbb{H}\)-valued random variables with emphasis on the main theorem of his Ph. D. thesis of 1991, the Marcinkiewicz-Zygmund law of large numbers, non-classical laws of the iterated logarithm [mainly following the work of H.-P. Scheffler, Publ. Math. 47, No. 3/4, 377-391 (1995; Zbl 0861.60015) and Stat. Probab. Lett. 24, No. 3, 187-192 (1995; Zbl 0832.60011)], and the two-series theorem.

The book adds favorably to the literature on recent advances in probability theory on groups. It is clearly written with an elaborate introduction and a very useful first chapter on basic notions and technical preparations of the method. The rich bibliography of 176 items goes far beyond the references actually employed in the text. As for the choice of the material treated in the book, the author aimed at describing the recent state of the art by orienting himself at his own contributions. Fortunately, he did not try to update the pioneering monographs of the 60s; such an enterprise would have exceeded the habitual size of a “Habilitationsschrift” and would certainly have spoiled the attractivity immanent in this valuable handy volume that will doubtlessly encourage further research interest in probability theory on groups.

Reviewer: H.Heyer (Tübingen)

### MSC:

60B15 | Probability measures on groups or semigroups, Fourier transforms, factorization |

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

22E25 | Nilpotent and solvable Lie groups |

47D06 | One-parameter semigroups and linear evolution equations |

60Fxx | Limit theorems in probability theory |