##
**Decoupling. From dependence to independence. Randomly stopped processes, \(U\)-statistics and processes, martingales and beyond.**
*(English)*
Zbl 0918.60021

New York, NY: Springer. xv, 392 p. (1998).

There are expressions involving dependent random variables that have a clear decoupled analogue, that is, a similar expression where some of the dependence has been replaced by conditional independence. For example, if \(\{X_i\}\) are independent, \(T\) is a stopping time for the \(X\)-sequence and \(\{\widetilde X_i\}\) is an independent copy of \(\{X_i\}\), then the decoupled analogue of the random sum \(\sum^T_{i=1}X_i\) is \(\sum^T_{i=1} \widetilde X_i\), which, conditionally on the \(X\) sequence, is a non-random sum of independent random variables, and is easier to analyze; for \(U\)-statistics \(\sum_{i\neq j} h(X_i,X_j)\) the decoupled analogue is \(\sum_{i\neq j} h(X_i, \widetilde X_j)\), where the different entries of \(h\) are filled up with random variables from different independent sequences, so that, conditionally on all but one entry, the \(U\)-statistic becomes a sum of independent random variables; a third relevant example, which was historically the first, corresponds to martingale transforms. There is a general framework that encompasses all these examples, namely tangent sequences. Then, a decoupling inequality is an inequality relating probabilistic characteristics of an expression and its decoupled, mainly moments, weak moments, exponential moments and tail probabilities.

This book gives an account of decoupling inequalities and some of their applications. It begins with a chapter on sums of independent random vectors and variables that includes very recent material such as, among other results, Khinchin’s inequality for Rademacher vectors with best constants, the Montgomery-Smith maximal inequality for the partial sums of an i.i.d. sequence, Latala’s sharp bounds for moments of sums of independent variables, and Rosenthal’s and Hoffmann-Jørgensen’s inequalities with sharp constants; as an application, the first decoupling inequality in the book, comparing the \(L_p\) norms of arbitrary positive random variables with those of sums of independent random variables with the same marginals, due to de la Peña, is obtained. The results of this chapter are used in the sequel. Then, decoupling is introduced by means of two of the main examples, sums of a random number of independent random vectors, with applications in boundary crossing by processes with independent increments (Chapter 2) and \(U\)-statistics (Chapter 3). The best decoupling results for these two categories of examples cannot be deduced from the general theory, developed later in the book. Chapters 4 and 5 are devoted to the asymptotic theory of \(U\)-statistics and \(U\)-processes, whose recent developments – exponential inequalities, limit theorems – constitute one of the main successes of decoupling; hypercontractivity of Rademacher and Gaussian chaos, a construction of the Gaussian chaos in connection with the central limit theorem, and general weak convergence of processes are aside subjects developed in detail in order to make these two chapters self-contained. Chapter 5 also contains some statistical applications of \(U\)-processes.

The last three chapters are devoted to the general theory of decoupling and its interplay with martingale theory. Chapter 6 and 7 develop decoupling in the general context of tangent sequences: \(L_p\) and good-lambda inequalities, differential subordination, comparison between decoupling and martingale inequalities, the principle of conditioning. Among other applications, the sharper decoupling inequalities allow almost direct transfer of properties of sums of independent random variables to martingales as illustrated with a proof of the Burkholder extension of Rosenthal’s inequality with constants of the right order of magnitude. The last chapter contains further applications, notably the extension of Wald’s equations to randomly stopped \(U\)-statistics, and exponential inequalities for the ratios between martingales and their conditional variances obtained, via decoupling, from the classical exponential inequalities for sums of independent random variables.

A large portion of the material, developed in recent years by many researchers, including the authors, is presented in book form for the first time. The book is addressed to researchers and graduate students in probability and statistics and the exposition is at the level of a second graduate probability course (with parts that can be used in a first year course).

This book gives an account of decoupling inequalities and some of their applications. It begins with a chapter on sums of independent random vectors and variables that includes very recent material such as, among other results, Khinchin’s inequality for Rademacher vectors with best constants, the Montgomery-Smith maximal inequality for the partial sums of an i.i.d. sequence, Latala’s sharp bounds for moments of sums of independent variables, and Rosenthal’s and Hoffmann-Jørgensen’s inequalities with sharp constants; as an application, the first decoupling inequality in the book, comparing the \(L_p\) norms of arbitrary positive random variables with those of sums of independent random variables with the same marginals, due to de la Peña, is obtained. The results of this chapter are used in the sequel. Then, decoupling is introduced by means of two of the main examples, sums of a random number of independent random vectors, with applications in boundary crossing by processes with independent increments (Chapter 2) and \(U\)-statistics (Chapter 3). The best decoupling results for these two categories of examples cannot be deduced from the general theory, developed later in the book. Chapters 4 and 5 are devoted to the asymptotic theory of \(U\)-statistics and \(U\)-processes, whose recent developments – exponential inequalities, limit theorems – constitute one of the main successes of decoupling; hypercontractivity of Rademacher and Gaussian chaos, a construction of the Gaussian chaos in connection with the central limit theorem, and general weak convergence of processes are aside subjects developed in detail in order to make these two chapters self-contained. Chapter 5 also contains some statistical applications of \(U\)-processes.

The last three chapters are devoted to the general theory of decoupling and its interplay with martingale theory. Chapter 6 and 7 develop decoupling in the general context of tangent sequences: \(L_p\) and good-lambda inequalities, differential subordination, comparison between decoupling and martingale inequalities, the principle of conditioning. Among other applications, the sharper decoupling inequalities allow almost direct transfer of properties of sums of independent random variables to martingales as illustrated with a proof of the Burkholder extension of Rosenthal’s inequality with constants of the right order of magnitude. The last chapter contains further applications, notably the extension of Wald’s equations to randomly stopped \(U\)-statistics, and exponential inequalities for the ratios between martingales and their conditional variances obtained, via decoupling, from the classical exponential inequalities for sums of independent random variables.

A large portion of the material, developed in recent years by many researchers, including the authors, is presented in book form for the first time. The book is addressed to researchers and graduate students in probability and statistics and the exposition is at the level of a second graduate probability course (with parts that can be used in a first year course).

Reviewer: E.Gine (Storrs)

### MSC:

60E15 | Inequalities; stochastic orderings |

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

60G42 | Martingales with discrete parameter |

60G50 | Sums of independent random variables; random walks |

60G40 | Stopping times; optimal stopping problems; gambling theory |

60F05 | Central limit and other weak theorems |

60F17 | Functional limit theorems; invariance principles |

60J99 | Markov processes |

60F15 | Strong limit theorems |

62E20 | Asymptotic distribution theory in statistics |