Probabilistic limit theorems in the setting of Banach spaces.

*(English)*Zbl 1039.60003
Johnson, W. B. (ed.) et al., Handbook of the geometry of Banach spaces. Volume 2. Amsterdam: North-Holland (ISBN 0-444-51305-1/hbk). 1177-1200 (2003).

Developments of Banach space techniques in the context of classical limit theorems in probability theory are given. Let \(B\) be a real separable Banach space with norm \(\|\cdot\|\) and topological dual space \(B^\prime\). Let \(X\) be a Borel random variable on some probability space \((\Omega, \mathcal A, \mathcal P)\) with values in \(B\), and let \((X_n)_{n\in\mathcal N}\) be a sequence of independent copies of \(X\). For each \(n\geq1\), set \(S_n=X_1+\cdots+X_n\). The classical probability theory on \(\mathbb R\) or \(\mathbb R^k\) is mostly concerned with the limiting behaviour of the partial sum sequence \((S_n)_{n\geq1}\). The most important and famous results are the (strong) law of large numbers (LLN), the central limit theorem (CLT) and the law of the iterated logarithm (LIL). For convenience, the authors mostly refer to the monograph by M. Ledoux and M. Talagrand [“Probability in Banach spaces. Isoperimetry and processes” (1991; Zbl 0748.60004)] for a complete account on the subject of probability in Banach spaces, for further references and historical developments and for the complete proofs that are only outlined in this survey.

Let \(X\) be a real-valued random variable. It is well-known that:

(a) The sequence \((S_n/n)_{n\geq1}\) converges almost surely (a.s.) to \(E(X)\) if and only if \(E(| X| )<\infty\) (we then say that \(X\) satisfies the LLN, or strong LLN).

(b) The sequence \((S_n/n)_{n\geq1}\) converges in distribution (to a normal random variable \(G\)) if and only if \(E(X)=0\) and \(\sigma^2= E(X^2)< \infty\) (and in this case, \(G\) is centered with variance \(\sigma^2\)) (we then say that \(X\) satisfies CLT).

(c) Define on \(\mathbb R^+\) the function \(LLx=\log\log x\) if \(x\geq e\), and \(LLx=1\) if \(x<e\), and set \(a_n=(2nLLn)^{1/2}\) for every \(n\geq1\). The sequence \((S_n/n)_{n\geq1}\) is a.s. bounded if and only if \(E(x)=0\) and \(\sigma^2=E(X^2)<\infty\), and in this case \({\lim\inf}_{n\to\infty}{S_n\over a_n}=-\sigma\) and \({\lim\sup}_{n\to\infty}{S_n\over a_n}=+\sigma\) with probability 1. Moreover, the set of limit points of the sequence \((S_n/n)_{n\geq1}\) is a.s. equal to the interval \([-\sigma,+\sigma]\) (we then say that \(X\) satisfies the LIL).

The definitions of these basic limit theorems extend to random variables taking values in an infinite-dimensional real separable Banach space \(B\). Weak convergence in the CLT has to be understood as weak convergence in the space of Borel probability measures on the complete separable metric space \(B\). For the LIL, one has to distinguish between a bounded form and a compact form. Moment conditions on the law of \(X\) fully characterize the preceding limit theorems in finite dimension. However this is no longer true in infinite dimension. At this point emphasis was put on understanding what kind of conditions on the space can ensure an extension of the finite-dimensional statements and what new descriptions are available in this setting.

In the first part of this survey the a.s. limit theorems (LLN and LIL) are described. As a main observation, it is established, as a consequence of deep exponential bounds, which are parts of the concentration of measure phenomenon for product measures, that the a.s. statements actually reduce to the corresponding ones in probability or in distribution under necessary moment conditions. It is a main contribution of the Banach space approach to realize that moment conditions are actually used to handle convergence in distribution. This fact is further illustrated in Section 3 in the investigation of the classical CLT using type and cotype. In the last paragraph applications of these ideas and techniques to empirical processes and bootstrap in statistics are described.

For the entire collection see [Zbl 1013.46001].

Let \(X\) be a real-valued random variable. It is well-known that:

(a) The sequence \((S_n/n)_{n\geq1}\) converges almost surely (a.s.) to \(E(X)\) if and only if \(E(| X| )<\infty\) (we then say that \(X\) satisfies the LLN, or strong LLN).

(b) The sequence \((S_n/n)_{n\geq1}\) converges in distribution (to a normal random variable \(G\)) if and only if \(E(X)=0\) and \(\sigma^2= E(X^2)< \infty\) (and in this case, \(G\) is centered with variance \(\sigma^2\)) (we then say that \(X\) satisfies CLT).

(c) Define on \(\mathbb R^+\) the function \(LLx=\log\log x\) if \(x\geq e\), and \(LLx=1\) if \(x<e\), and set \(a_n=(2nLLn)^{1/2}\) for every \(n\geq1\). The sequence \((S_n/n)_{n\geq1}\) is a.s. bounded if and only if \(E(x)=0\) and \(\sigma^2=E(X^2)<\infty\), and in this case \({\lim\inf}_{n\to\infty}{S_n\over a_n}=-\sigma\) and \({\lim\sup}_{n\to\infty}{S_n\over a_n}=+\sigma\) with probability 1. Moreover, the set of limit points of the sequence \((S_n/n)_{n\geq1}\) is a.s. equal to the interval \([-\sigma,+\sigma]\) (we then say that \(X\) satisfies the LIL).

The definitions of these basic limit theorems extend to random variables taking values in an infinite-dimensional real separable Banach space \(B\). Weak convergence in the CLT has to be understood as weak convergence in the space of Borel probability measures on the complete separable metric space \(B\). For the LIL, one has to distinguish between a bounded form and a compact form. Moment conditions on the law of \(X\) fully characterize the preceding limit theorems in finite dimension. However this is no longer true in infinite dimension. At this point emphasis was put on understanding what kind of conditions on the space can ensure an extension of the finite-dimensional statements and what new descriptions are available in this setting.

In the first part of this survey the a.s. limit theorems (LLN and LIL) are described. As a main observation, it is established, as a consequence of deep exponential bounds, which are parts of the concentration of measure phenomenon for product measures, that the a.s. statements actually reduce to the corresponding ones in probability or in distribution under necessary moment conditions. It is a main contribution of the Banach space approach to realize that moment conditions are actually used to handle convergence in distribution. This fact is further illustrated in Section 3 in the investigation of the classical CLT using type and cotype. In the last paragraph applications of these ideas and techniques to empirical processes and bootstrap in statistics are described.

For the entire collection see [Zbl 1013.46001].

Reviewer: Vakhtang V. Kvaratskhelia (Tbilisi)

##### MSC:

60B12 | Limit theorems for vector-valued random variables (infinite-dimensional case) |

46B09 | Probabilistic methods in Banach space theory |

46-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to functional analysis |

60-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to probability theory |