摘要:

This essay gives an outline of some of Wim Vervaat's innovative research ideas, especially those involving sup measures and capacities. These ideas were developed at his initiative in order to find satisfactory solutions to some limit problems in probability theory. The essay discusses the motivating questions and the role of sup measures and capacities in their solutions.

ISSN:0039-0402    

DOI:10.1111/j.1467-9574.1996.tb01491.x

出版商:Blackwell Publishing Ltd    

年代:1996

数据来源: WILEY

摘要:

Let (Xn, n ≥1) be an i.i.d. sequence of positive random variables with distribution functionH. Let φH:={(n, Xn), n ≥ 1) be the associated observation process. We view φhas a measure onE:= [0, ∞) ∞ (0, φ] where φH(A) is the number of points of φHwhich lie inA. A family (Vs, s>0) of transformations is defined onEin such a way that for suitableHthe distributions of (VsφH, S>0) satisfy a large deviation principle and that a related Strassen‐type law of the iterated logarithm also holds. Some consequent large deviation principles and loglog laws are derived for extreme values. Similar results are proved for φHreplaced by certain planar P

ISSN:0039-0402    

DOI:10.1111/j.1467-9574.1996.tb01492.x

出版商:Blackwell Publishing Ltd    

年代:1996

数据来源: WILEY

摘要:

LetF, denote the uniform empirical distribution based on the firstn ≥1 observations from an i.i.d. sequence of uniform (0, 1) random variables. We describe the almost sure limiting behavior of the sets of increment functions{Fn(t + hn.) ‐ Fn(t):0 ≤t ≤1 ‐ hn}, when{hn: n ≥1) is a nonincreasing sequence of constants such that

ISSN:0039-0402    

DOI:10.1111/j.1467-9574.1996.tb01493.x

出版商:Blackwell Publishing Ltd    

年代:1996

数据来源: WILEY

摘要:

We introduce a new method for proving large‐deviation principles (LDPs). This method amounts to “mixing” a collection of LDPs with a sequence of probability measures that obeys itself an LDP. As an illustration, we construct from empirical measures a sequence of capacity‐valued maps that can be considered as an indexed collection of LDPs. The index 1 coincides with Sanov's theorem. By taking a Poisson mixture we establish a new connection with Cramér's

ISSN:0039-0402    

DOI:10.1111/j.1467-9574.1996.tb01494.x

出版商:Blackwell Publishing Ltd    

年代:1996

数据来源: WILEY

摘要:

A random variableXon IR+is said to be self‐decomposable, dif for all c∈ (0, 1) there exists a random variableXcon IR+such thatX=dcX+Xc. It is said to be stable if it is self‐decomposable andXc=d(1 ‐c)X', whereXandX'are identically and independently distributed. The notions of stability and self‐decomposability for infinitely divisible random variables are generalised to abelian semi‐groups (S, +) with S having an identical involution, by using characteristic functions. The generalised definitions involve semi‐groups of scaling operatorsT. There operators can be interpreted in a slightly different context as generalised continuous‐time branching processes (with immigration). The underlying importance of the generator of the semi‐groupsTin the characterisation of stability and self‐decompos

ISSN:0039-0402    

DOI:10.1111/j.1467-9574.1996.tb01495.x

出版商:Blackwell Publishing Ltd    

年代:1996

数据来源: WILEY

摘要:

Capacities are defined as set functions with regularity properties in terms of two general familiesKandGof ‘inner’ and ‘outer’ sets (K =compact sets,G= open sets in the standard case). The spaceCof all such capacities is a complete lattice with respect to its natural partial order. The lattice structure ofCis studied in combination with the natural topology ofC. Also some subspaces are inves

ISSN:0039-0402    

DOI:10.1111/j.1467-9574.1996.tb01496.x

出版商:Blackwell Publishing Ltd    

年代:1996

数据来源: WILEY

ISSN:0039-0402    

DOI:10.1111/j.1467-9574.1996.tb01497.x

出版商:Blackwell Publishing Ltd    

年代:1996

数据来源: WILEY