A note on the von Weizsäcker theorem
Introduction
Komlós’s theorem states that every -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 -boundedness can be dropped. Its main result states that every nonnegative sequence in 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 be a sequence of nonnegative random variables. Then there exist a subsequence and a nonnegative random variable such that the following statements are true: The subsequence is -almost surely Cesàro convergent to . There exists an equivalent probability measure such that the sequence is -bounded.
In addition, we can choose the subsequence such that even one of the following stronger conditions is fulfilled:
- (a)
For every permutation the sequence is -almost surely Cesàro convergent to .
- (a)
For every further subsequence the sequence is -almost surely Cesàro convergent to .
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 be a subsequence as in Theorem 1.1; that is, conditions (a) and (b) are fulfilled. Note that the limit can be infinite. In this note we will provide a description of the sets and . In von Weizsäcker (2004) it is already indicated that should be the largest subset on which is -bounded for some equivalent probability measure . 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 there exists a partition such that is bounded in probability and is hereditarily unbounded in probability on . We refer to Section 2 for the precise definitions. The set is characterized as the largest subset on which the convex set is bounded in probability.
Now, we define the convex hulls as where and denote by and the corresponding partitions according to the decomposition result from Brannath and Schachermayer (1999). Note that , because Our result reads as follows.
Proposition 1.2 We have and up to -null sets.
Remark 1.3 Let us mention some further consequences: The set is the largest subset on which the convex hull of or is bounded in probability. The set is also the largest subset on which the sequence or is -bounded for some equivalent probability measure . This is in accordance with the findings in von Weizsäcker (2004). The set is also the largest subset on which every sequence of convex combinations of or 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 is almost surely finite. Note that Proposition 1.2 provides a link between the set and the partition , which can be used in two directions; in particular, in certain situations we can conclude that the sets and are bounded in probability. Suppose we have given a subsequence as in the von Weizsäcker theorem (Theorem 1.1), and suppose we know the partition . Then we can easily determine the set ; this is illustrated in Section 3 in the situation where we have an atomic probability space. Conversely, suppose we have given a subsequence as in the von Weizsäcker theorem (Theorem 1.1), and suppose we know the set . Then we can easily determine the partitions and . For illustration, we will assume in Section 4 that the sequence satisfies the strong law of large numbers (SLLN). As we will see, then we can take the original sequence in the von Weizsäcker theorem; that is, we do not have to pass to a subsequence . As a consequence, the sets and are given by and we will provide a criterion when these sets are bounded in probability.
Section snippets
Proof of the result
Let be a probability space. We denote by the space of all equivalence classes of random variables, where two random variables and are identified if . We denote by the convex cone of all nonnegative random variables; that is, all such that . It is well-known that equipped with the translation invariant metric is 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 is atomic. More precisely, we assume there are subsets of such that the following conditions are fulfilled:
- •
The sets are pairwise disjoint with .
- •
We have .
- •
We have for all .
Let be a sequence of nonnegative random variables. Furthermore, let be a subsequence and let 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 be a sequence of nonnegative random variables. We say that satisfies the strong law of large numbers (SLLN) if there is a nonnegative, possibly infinite constant such that the following conditions are fulfilled:
- (1)
We have for all .
- (2)
is -almost surely Cesàro convergent to .
If the sequence 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)
On the laws of large numbers for nonnegative random variables
J. Multivariate Anal.
(1983)An extension of the Komlós subsequence theorem
Acta Math. Hungar.
(1990)- et al.
A bipolar theorem for
Kolmogoroff’s strong law of large numbers for pairwise uncorrelated random variables
(2020)
Cited by (7)
Permutation-invariance in Komlós type theorem for non-negative random variables
2023, Indagationes MathematicaeA strong law of large numbers for positive random variables
2023, Illinois Journal of MathematicsNo-arbitrage concepts in topological vector lattices
2021, Positivity