Some limit theorems for empirical processes (with discussion).

*(English)*Zbl 0553.60037The invited paper on the CLT for empirical processes indexed by classes of functions starts with a neat description of the basic tools of this field. Among others it introduces functional P-Donsker classes, random entropies, metric entropies, Vapnik-Červonenkis classes as well as K. Alexander’s measurability concepts and adaptions of relevant inequalities from probability theory in Banach spaces.

The following sections 3 to 8 are concerned with conditions which ensure the CLT for empirical processes indexed by functions, which modulo some measurability conditions ensure that the function class is a P-Donsker class [using a result of R. M. Dudley and W. Philipp, Z. Wahrscheinlichkeitstheor. Verw. Geb. 62, 509-552 (1983; Zbl 0488.60044)]. One interesting result for uniformly bounded function classes is that a P-Donsker class is a P-pregaussian class satisfying some ’reduced’ randomized equicontinuity condition. The main result of the paper (Theorem 5.7) shows (using an idea of Le Cam) that the equicontinuity condition may be replaced by some tail condition for random entropies for the case of set classes.

Applications of these interesting results are made to uniformly bounded processes in C[0,1] and D[0,1] as well as to classes of unbounded functions. Some of the technical lemmas of this paper might be useful in other contexts than the CLT.

The paper is followed by a discussion in which K. Alexander emphasizes the usefulness of Le Cam’s square root trick’ (Lemma 5.2), R. M. Dudley illuminates the relation with the extensive literature on the CLT in Banach spaces, whereas P. Gaenssler points out application in non parametric statistics estimating spatial patterns. W. Philipp mentions the connection with the law of the iterated logarithm, D. Pollard comments on various inequalities, R. Pyke emphasizes unsolved problems in connection with computer aided sampling and W. Stute comments on the fact the paper of Giné and Zinn does not apply directly to the invariance principle for the empirical characteristic function.

The following sections 3 to 8 are concerned with conditions which ensure the CLT for empirical processes indexed by functions, which modulo some measurability conditions ensure that the function class is a P-Donsker class [using a result of R. M. Dudley and W. Philipp, Z. Wahrscheinlichkeitstheor. Verw. Geb. 62, 509-552 (1983; Zbl 0488.60044)]. One interesting result for uniformly bounded function classes is that a P-Donsker class is a P-pregaussian class satisfying some ’reduced’ randomized equicontinuity condition. The main result of the paper (Theorem 5.7) shows (using an idea of Le Cam) that the equicontinuity condition may be replaced by some tail condition for random entropies for the case of set classes.

Applications of these interesting results are made to uniformly bounded processes in C[0,1] and D[0,1] as well as to classes of unbounded functions. Some of the technical lemmas of this paper might be useful in other contexts than the CLT.

The paper is followed by a discussion in which K. Alexander emphasizes the usefulness of Le Cam’s square root trick’ (Lemma 5.2), R. M. Dudley illuminates the relation with the extensive literature on the CLT in Banach spaces, whereas P. Gaenssler points out application in non parametric statistics estimating spatial patterns. W. Philipp mentions the connection with the law of the iterated logarithm, D. Pollard comments on various inequalities, R. Pyke emphasizes unsolved problems in connection with computer aided sampling and W. Stute comments on the fact the paper of Giné and Zinn does not apply directly to the invariance principle for the empirical characteristic function.

Reviewer: F.Götze

##### MSC:

60F17 | Functional limit theorems; invariance principles |

60F05 | Central limit and other weak theorems |

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

62E20 | Asymptotic distribution theory in statistics |