Central limit theorems for generalized multilinear forms.

*(English)*Zbl 0677.60029
CWI Tracts, 61. Amsterdam: Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica. 84 p. Dfl. 14.10 (1989).

This monograph gives a clear and carefully argued analysis of conditions under which generalized multilinear forms converge to a normal limit. These results are an extension of the results of the author [Probab. Theory Relat. Fields 75, 261-277 (1987; Zbl 0596.60022)].

The first chapter provides an introduction, brief overview of the key results and a survey of the methods used by others to obtain results on multilinear forms. In Chapter 2, ‘clean’ random variables are defined and the main result is stated and proved. Consider the families of random variables \({\mathcal W}_ n=\{W_ I:\) \(I\in \{1,2,...,n\}\}\), indexed by finite subsets of the integers, and the \(\sigma\)-algebras \({\mathcal F}_ n^{(i)}=\sigma \{W_ I\in {\mathcal W}_ n:\) \(i\not\in I\}\). The random variable \(W_ I\) is ‘clean’ if \(E(W_ I| {\mathcal F}_ n^{(i)})=0\) a.s. for all \(i\in I.\)

The main result is a central limit theorem for \(W(n)=\sum_{| I| =d}W_ I\), homogeneous sums of degree d of clean random variables. Examples of such sums are the components in the Hoeffding decomposition for a U-statistic. The text includes a thorough discussion of the role of the conditions in each of the theorems and examples are included to illustrate the need for certain conditions.

Chapter 3 starts with a technical discussion of alternative conditions which can be used to ensure the asymptotic normality of W(n). In particular, the conditions are simplified for homogeneous sums in the Hoeffding decomposition. This is the case where there is an underlying sequence of independent random variables \(X_ 1,X_ 2,...\), \({\mathcal F}_ I=\sigma \{X_ i:\) \(i\in I\}\), \(W_ I=w_{nI}(X_{i_ 1},...,X_{i_ d})\) for \(I=(i_ 1,...,i_ d)\) with \(w_{nI}\) real- valued Borel measurable functions, and \(E(W_ I| {\mathcal F}_ J)=0\) a.s. if \(I\setminus J\neq \emptyset\). In the last section of Chapter 3, the results on homogeneous sums are applied to obtain central limit theorems for inhomogeneous sums of the form \(V(n)=\sum^{d}_{e=1}\sum_{| I| =d}W_ I.\)

The final chapter returns to homogeneous sums of degree d in the Hoeffding decomposition. The main aim of this chapter is to show how the random variables \(W_ I\) can be embedded as random point masses at points \(x_ I\) in a suitable product space \(E^ d\). Given this embedding, a class of functions f is identified such that the stochastic integral \(\int f dW(n)=\sum_{| I| =d}f(x_ I)W_ I\) converges to a stochastic integral with respect to a Gaussian process with independent increments. As a preliminary, conditions are given under which the distributions of the partial sums of the components of W(n) converge to the normal distribution.

The first chapter provides an introduction, brief overview of the key results and a survey of the methods used by others to obtain results on multilinear forms. In Chapter 2, ‘clean’ random variables are defined and the main result is stated and proved. Consider the families of random variables \({\mathcal W}_ n=\{W_ I:\) \(I\in \{1,2,...,n\}\}\), indexed by finite subsets of the integers, and the \(\sigma\)-algebras \({\mathcal F}_ n^{(i)}=\sigma \{W_ I\in {\mathcal W}_ n:\) \(i\not\in I\}\). The random variable \(W_ I\) is ‘clean’ if \(E(W_ I| {\mathcal F}_ n^{(i)})=0\) a.s. for all \(i\in I.\)

The main result is a central limit theorem for \(W(n)=\sum_{| I| =d}W_ I\), homogeneous sums of degree d of clean random variables. Examples of such sums are the components in the Hoeffding decomposition for a U-statistic. The text includes a thorough discussion of the role of the conditions in each of the theorems and examples are included to illustrate the need for certain conditions.

Chapter 3 starts with a technical discussion of alternative conditions which can be used to ensure the asymptotic normality of W(n). In particular, the conditions are simplified for homogeneous sums in the Hoeffding decomposition. This is the case where there is an underlying sequence of independent random variables \(X_ 1,X_ 2,...\), \({\mathcal F}_ I=\sigma \{X_ i:\) \(i\in I\}\), \(W_ I=w_{nI}(X_{i_ 1},...,X_{i_ d})\) for \(I=(i_ 1,...,i_ d)\) with \(w_{nI}\) real- valued Borel measurable functions, and \(E(W_ I| {\mathcal F}_ J)=0\) a.s. if \(I\setminus J\neq \emptyset\). In the last section of Chapter 3, the results on homogeneous sums are applied to obtain central limit theorems for inhomogeneous sums of the form \(V(n)=\sum^{d}_{e=1}\sum_{| I| =d}W_ I.\)

The final chapter returns to homogeneous sums of degree d in the Hoeffding decomposition. The main aim of this chapter is to show how the random variables \(W_ I\) can be embedded as random point masses at points \(x_ I\) in a suitable product space \(E^ d\). Given this embedding, a class of functions f is identified such that the stochastic integral \(\int f dW(n)=\sum_{| I| =d}f(x_ I)W_ I\) converges to a stochastic integral with respect to a Gaussian process with independent increments. As a preliminary, conditions are given under which the distributions of the partial sums of the components of W(n) converge to the normal distribution.

Reviewer: N.Weber

##### MSC:

60F05 | Central limit and other weak theorems |

60H05 | Stochastic integrals |

60G15 | Gaussian processes |