Money metric utilitarianism

We discuss a method of ranking allocations in economic environments which applies when we do not know the names or preferences of individual agents. We require that two allocations can be ranked with the knowledge only of their aggregate bundles and community indifference sets–a condition we refer to as aggregate independence. We also postulate a basic Pareto and continuity property, and a property stating that when two disjoint economies and allocations are put together, the ranking in the large economy should be consistent with the rankings in the two smaller economies (reinforcement). We show that a ranking method satisfies these axioms if and only if there is a probability measure over the strictly positive prices for which the rule ranks allocations on the basis of the random-price money-metric utilitarian rule. This is a rule which computes the money-metric utility for each agent at each price, sums these, and then takes an expectation according to the probability measure

Random walk boundaries: their entropies and connections with Hecke pairs

We present three papers in non-singular dynamics concerning boundaries of random walks on locally compact, second countable groups. One common theme is entropy. Paper II and III are concerned with boundary entropy spectra, while Paper I studies topological properties of entropy. In Paper II we moreover establish a technique to relate random walks on locally profinite groups to random walks on dense discrete subgroups, by the concept of Hecke pairs, which is also used in Paper III.In Paper I we introduce different perspectives and extensions of Furstenberg\u27s entropy and show semi-continuity and continuity results in these contexts. In particular we apply these to upper and lower limits of non-nested sequences of sigma-algebras in the sense of Kudo.Paper II relates certain random walks on locally profinite groups to random walks on dense discrete subgroups, using a Hecke subgroup, such that the Poisson boundary of the first becomes a boundary of the second one. If the Poisson boundaries of these two walks happen to coincide, then the Hecke subgroup in charge has to be amenable. For some random walks on lamplighter and solvable Baumslag-Solitar groups we obtain that their Poisson boundary is prime and the quasi-regular representation is reducible. Moreover, we find a group such that for any given summable sequence of positive numbers there is a random walk whose boundary entropy spectrum equals the subsum set of this sequence. In particular we obtain a boundary entropy spectrum which is a Cantor set and one which is an interval.In Paper III we study the boundary entropy spectra of finitely supported, generating random walks on a certain affine group, realizing them as finite subsum sets. We show that the averaged information function of a stationary probability measure does not change when passing to a non-singular, absolutely continuous sigma-finite measure and deduce an entropy formula

THE POSSIBILITIES OF APPLYING HISTORIC SIMULATION FOR ASSESSING THE PROFITABILITY OF INVESTMENTS

The above methods of assessment are based on different methodological and mathematical approaches; therefore, they yield slightly different results, which can be used to complement each other and thus gain a more complex picture about the risk of the investment. Calculating break-even points and sensitivity assessment are simple variance calculation methods with differing frameworks or backgrounds. The basis of decision is NPV, as the break even points of the various factors do not come complete with clearly defined rules and cut-off points for accepting or rejecting an investment. This is especially true for the dynamic payback period, which discriminates against long-term investments. Conclusions about the risk of the investment can be drawn if expected values and critical values show high disparity; this can be used for highlighting the more sensitive risk factors. However, in this case, we do not have a distribution of probability for factors, nor the probability of the occurrence of the critical values, so the use of data is limited. It is worth starting risk assessment with these simple methods of analysis to gain more insight into the underlying processes. If historic simulation is applied, we gain data for analysis from past figures. The basis for this procedure is sufficient quantity and quality data, which might pose considerable difficulties. On one hand, in the case of Hungary, the recent economic changes break the continuity of the data; on the other hand, it is often almost impossible to collect the necessary minimum of 50-100 figures for a realistic simulation generating the distribution of NPV. Setting the optimal time framework is also crucial: if data reach far back in the past, their applicability for present risk assessment is questionable. If the framework spans too short a period, the representativeness of the figures may be doubted. A frequent criticism of the method questions the approach that using real data from the past implies that the events in the past are likely to repeat, and new patterns are not expected to emerge. On the other hand, an advantage of the method is that it can be considered a complex calculation of variance that enables the analyst to assess possible future scenarios without having to define each and every expectation and prospect for the future. A wide range of mathematical skills is not necessary; therefore, it is easy to use, and the results are readily applicable, and expand on the results gained from more simple calculations aimed at assessing the risk of investments. All the above methods make an attempt at grabbing and describing risk itself. However, they simply cannot consider that investors may have different approaches to risk, which has a crucial impact on the decision about the investment. Investors expect benefits and positive outcomes proportionate to the risk undertaken; as a result, they have different sets of priorities for future cash flow. Although this factor does not appear in any of the models, using multiple models for an analysis can provide a wider range of information to be considered for assessing the profitability of an investment.risk factors, risk assessment methods, simulation, Agricultural Finance, Financial Economics,

Continuity of Channel Parameters and Operations under Various DMC Topologies

We study the continuity of many channel parameters and operations under various topologies on the space of equivalent discrete memoryless channels (DMC). We show that mutual information, channel capacity, Bhattacharyya parameter, probability of error of a fixed code, and optimal probability of error for a given code rate and blocklength, are continuous under various DMC topologies. We also show that channel operations such as sums, products, interpolations, and Ar{\i}kan-style transformations are continuous.Comment: 31 pages. Submitted to IEEE Trans. Inform. Theory and in part to ISIT201

Continuity of the von Neumann entropy

A general method for proving continuity of the von Neumann entropy on subsets of positive trace-class operators is considered. This makes it possible to re-derive the known conditions for continuity of the entropy in more general forms and to obtain several new conditions. The method is based on a particular approximation of the von Neumann entropy by an increasing sequence of concave continuous unitary invariant functions defined using decompositions into finite rank operators. The existence of this approximation is a corollary of a general property of the set of quantum states as a convex topological space called the strong stability property. This is considered in the first part of the paper.Comment: 42 pages, the minor changes have been made, the new applications of the continuity condition have been added. To appear in Commun. Math. Phy

Asymptotics for relative frequency when population is driven by arbitrary evolution

Strongly consistent estimates are shown, via relative frequency, for the probability of "white balls" inside a dichotomous urn when such a probability is an arbitrary continuous time dependent function over a bounded time interval. The asymptotic behaviour of relative frequency is studied in a nonstationary context using a Riemann-Dini type theorem for SLLN of random variables with arbitrarily different expectations; furthermore the theoretical results concerning the SLLN can be applied for estimating the mean function of unknown form of a general nonstationary process.Comment: 29 page