A note on the von Weizsäcker theorem

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Abstract

The von Weizsäcker theorem states that every sequence of nonnegative random variables has a subsequence which is Cesàro convergent to a nonnegative random variable which might be infinite. The goal of this note is to provide a description of the set where the limit is finite. For this purpose, we use a decomposition result due to Brannath and Schachermayer.

Introduction

Komlós’s theorem states that every L1-bounded sequence has a subsequence which is Cesàro convergent to a finite limit; see Komlós (1967) and Berkes (1990). The paper von Weizsäcker (2004) was dealing with the question whether the L1-boundedness can be dropped. Its main result states that every nonnegative sequence in L0 has a subsequence which is Cesàro convergent, but the limit can be infinite. More precisely, we have the following result.

Theorem 1.1 von Weizsäcker

Let (ξn)nNL+0 be a sequence of nonnegative random variables. Then there exist a subsequence (ξnk)kN and a nonnegative random variable ξ:Ω[0,] such that the following statements are true:

  • (a)

    The subsequence (ξnk)kN is P-almost surely Cesàro convergent to ξ.

  • (b)

    There exists an equivalent probability measure QP such that the sequence (ξnk1{ξ<})kN is L1(Q)-bounded.

In addition, we can choose the subsequence (ξnk)kN such that even one of the following stronger conditions is fulfilled:

  • (a)

    For every permutation π:NN the sequence (ξnπ(k))kN is P-almost surely Cesàro convergent to ξ.

  • (a)

    For every further subsequence (nkl)lN the sequence (ξnkl)lN is P-almost surely Cesàro convergent to ξ.

The existence of a subsequence such that (a) and (b) are fulfilled follows from von Weizsäcker (2004). Slightly modifying its proof by using Komlós’s theorem from Komlós (1967) (rather than the version from Berkes, 1990) gives us the existence of a subsequence such that (a) and (b) are fulfilled; see also Kabanov and Safarian (2009, Thm. 5.2.3) for such a result.

Let (ξnk)kN be a subsequence as in Theorem 1.1; that is, conditions (a) and (b) are fulfilled. Note that the limit ξ can be infinite. In this note we will provide a description of the sets {ξ<} and {ξ=}. In von Weizsäcker (2004) it is already indicated that {ξ<} should be the largest subset on which (ξnk)kN is L1(Q)-bounded for some equivalent probability measure QP. However, à priori it is not clear whether such a set exists. We will approach this problem by looking at sets which are bounded in probability, without performing a measure change. For this purpose, we will use a decomposition result from Brannath and Schachermayer (1999) which states that for every convex subset CL+0 there exists a partition {Ωb,Ωu} such that C|Ωb is bounded in probability and C is hereditarily unbounded in probability on Ωu. We refer to Section 2 for the precise definitions. The set Ωb is characterized as the largest subset on which the convex set C is bounded in probability.

Now, we define the convex hulls C,C̄L+0 as Cconv{ξnk:kN}andC̄conv{ξ̄nk:kN},where ξ̄nk1kl=1kξnlfor each kN,and denote by {Ωb,Ωu} and {Ω̄b,Ω̄u} the corresponding partitions according to the decomposition result from Brannath and Schachermayer (1999). Note that C̄C, because {ξ̄nk:kN}C.Our result reads as follows.

Proposition 1.2

We have {ξ<}=Ωb=Ω̄b and {ξ=}=Ωu=Ω̄u up to P-null sets.

Remark 1.3

Let us mention some further consequences:

  • (1)

    The set {ξ<} is the largest subset on which the convex hull of (ξnk)kN or (ξ̄nk)kN is bounded in probability.

  • (2)

    The set {ξ<} is also the largest subset on which the sequence (ξnk)kN or (ξ̄nk)kN is L1(Q)-bounded for some equivalent probability measure QP. This is in accordance with the findings in von Weizsäcker (2004).

  • (3)

    The set {ξ<} is also the largest subset on which every sequence of convex combinations of (ξnk)kN or (ξ̄nk)kN has a convergent subsequence in the sense of weak convergence of their distributions.

For more details, we refer to Section 2, and in particular Corollary 2.12, where we investigate when the limit ξ is almost surely finite. Note that Proposition 1.2 provides a link between the set {ξ<} and the partition {Ωb,Ωu}, which can be used in two directions; in particular, in certain situations we can conclude that the sets C and C̄ are bounded in probability. Suppose we have given a subsequence (ξnk)kN as in the von Weizsäcker theorem (Theorem 1.1), and suppose we know the partition {Ωb,Ωu}. Then we can easily determine the set {ξ<}; this is illustrated in Section 3 in the situation where we have an atomic probability space. Conversely, suppose we have given a subsequence (ξnk)kN as in the von Weizsäcker theorem (Theorem 1.1), and suppose we know the set {ξ<}. Then we can easily determine the partitions {Ωb,Ωu} and {Ω̄b,Ω̄u}. For illustration, we will assume in Section 4 that the sequence (ξn)nN satisfies the strong law of large numbers (SLLN). As we will see, then we can take the original sequence (ξn)nN in the von Weizsäcker theorem; that is, we do not have to pass to a subsequence (ξnk)kN. As a consequence, the sets C and C̄ are given by C=conv{ξn:nN}andC̄=conv{ξ̄n:nN},and we will provide a criterion when these sets are bounded in probability.

Section snippets

Proof of the result

Let (Ω,,P) be a probability space. We denote by L0=L0(Ω,,P) the space of all equivalence classes of random variables, where two random variables X and Y are identified if P(X=Y)=1. We denote by L+0=L+0(Ω,,P) the convex cone of all nonnegative random variables; that is, all XL0 such that P(X0)=1. It is well-known that L0 equipped with the translation invariant metric d(X,Y)=E[|XY|1],X,YL0is a complete topological vector space. The induced convergence is just convergence in probability;

Sequences on an atomic probability space

In this section we assume that the probability space (Ω,,P) is atomic. More precisely, we assume there are subsets (Am)mN of Ω such that the following conditions are fulfilled:

  • The sets (Am)mN are pairwise disjoint with Ω=mNAm.

  • We have =σ(Am:mN).

  • We have P(Am)>0 for all mN.

Let (ξn)nNL+0 be a sequence of nonnegative random variables. Furthermore, let (ξnk)kN be a subsequence and let ξ:Ω[0,] be a nonnegative random variable such that conditions (a) and (b) in the von Weizsäcker theorem (

Sequences satisfying the strong law of large numbers

As in the previous sections, let (ξn)nNL+0 be a sequence of nonnegative random variables. We say that (ξn)nN satisfies the strong law of large numbers (SLLN) if there is a nonnegative, possibly infinite constant μ[0,] such that the following conditions are fulfilled:

  • (1)

    We have E[ξn]=μ for all nN.

  • (2)

    (ξn)nN is P-almost surely Cesàro convergent to μ.

If the sequence (ξn)nN satisfies the SLLN, then conditions (a) and (b) in the von Weizsäcker theorem (Theorem 1.1) are fulfilled by choosing the

Acknowledgment

I am grateful to an anonymous referee for helpful comments and suggestions.

References (9)

  • EtemadiN.

    On the laws of large numbers for nonnegative random variables

    J. Multivariate Anal.

    (1983)
  • BerkesI.

    An extension of the Komlós subsequence theorem

    Acta Math. Hungar.

    (1990)
  • BrannathW. et al.

    A bipolar theorem for L+0(Ω,,P)

  • JanischM.

    Kolmogoroff’s strong law of large numbers for pairwise uncorrelated random variables

    (2020)
There are more references available in the full text version of this article.
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