Aggregation of infinite sequences
Introduction
The aggregation of a fixed finite number n of inputs from a given scale (chain of possible values) E is proceeded by means of some n-ary aggregation function which is non-decreasing in each argument and for each there are such thatObserve that if E is a bounded chain with top element and bottom element , then (1) is equivalent to the idempotency of and , i.e., and , compare [12], [14]. In the case of a finite but not a priori fixed number of inputs extended aggregation functions are applied, where for each , is an n-ary aggregation function, and is the identity function, see e.g. [5], [15]. The aim of this paper is to discuss the aggregation of an infinite sequence of inputs, i.e., of non-decreasing functions such that (1) (with ) is fulfilled. Obviously, such functions can be defined arbitrarily without any relation to the extended aggregation functions . Throughout this paper, we will deal with the most common continuous scale . Some other types of scales will be briefly mentioned in the concluding remarks.
The aggregation of infinitely but still countably many inputs is important in several mathematical areas, such as discrete probability theory, but also in non-mathematical areas, such as decision problems with an infinite jury, game theory with infinitely many players, etc. Though these theoretical tasks seem to be far from reality, they enable a better understanding of decision problems with an extremely huge jury, game theoretical problems with extremely many players, etc., see [9], [20], [22], [25].
We consider the set of all sequences , where . The input space equipped with the standard Cartesian ordering, i.e., means , is a complete lattice with bottom element and top element . We equip with the coordinatewise convergence, i.e., a sequence from converges to if and only if for all .
In Section 2, we will discuss infinitary aggregation functions on sequences possessing some a priori given properties, such as additivity, comonotone additivity, symmetry, etc. On the other side, in Section 3 we will discuss infinitary aggregation functions related to a given extended aggregation function , where special attention is taken on triangular norms, triangular conorms and weighted means. Note that the discussion of the infinitary arithmetic mean can be found in [10], [11].
Section snippets
Infinitary aggregation functions and their properties
We extend the notion of the usual aggregation function to the case of infinite inputs. Definition 1 A function is an (infinitary) aggregation function if it satisfies the following conditions: It is non-decreasing, i.e., implies . and .
Properties of these functions are defined similarly as the corresponding properties of n-ary aggregation functions, see e.g. [5], [12], [14]. Definition 2 An aggregation function is said to be additive, comonotone additive,
Extended aggregation functions and infinitary aggregation functions
For a given extended aggregation function , we will look for an appropriate aggregation function somehow linked to . A genuine approach is to define as a limit of If this limit exists, for any , we will accept given by (4) as an extension of to the domain and we will keep the notation also for whenever appropriate. The aggregation function will be called countably extendable.
A sufficient
Concluding remarks
Results of this paper are related to problems discussed, e.g., in [3]. Moreover, they can be applied to extend the results presented on finite domains, such as those in [18], [19], [24].
We have considered and discussed infinitary extensions of aggregation functions acting on . Completely analogous considerations can be done on arbitrary closed (extended) real scale . In the case when E is not a closed real interval, the possible range of defined on should be extended to
Acknowledgements
Both authors want to thank for a support of the bilatelar Slovakia-Serbia Project. The work of the first author was supported by grants APVV-0375-06 and VEGA1/4209/07. The second author would like to thank for the support in part by the project MNTRS 144012, Grant of MTA of HTMT, French-Serbian Project”Pavle Savić”, and by the Project ”Mathematical Models for Decision Making under Uncertain Conditions and Their Applications” of the Academy of Sciences and Arts of Vojvodina supported by the
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