Savage's Theorem Research Articles

De Finetti-type theorems on quasi-local algebras and infinite Fermi tensor products

Local actions of [Formula: see text], the group of finite permutations on [Formula: see text], on quasi-local algebras are defined and proved to be [Formula: see text]-abelian. It turns out that invariant states under local actions are automatically even, and extreme invariant states are strongly clustering. Tail algebras of invariant states are shown to obey a form of the Hewitt and Savage theorem, in that they coincide with the fixed-point von Neumann algebra. Infinite graded tensor products of [Formula: see text]-algebras, which include the CAR algebra, are then addressed as particular examples of quasi-local algebras acted upon [Formula: see text] in a natural way. Extreme invariant states are characterized as infinite products of a single even state, and a de Finetti theorem is established. Finally, infinite products of factorial even states are shown to be factorial by applying a twisted version of the tensor product commutation theorem, which is also derived here.

Convex geometry of finite exchangeable laws and de Finetti style representation with universal correlated corrections

We present a novel analogue for finite exchangeable sequences of the de Finetti, Hewitt and Savage theorem and investigate its implications for multi-marginal optimal transport (MMOT) and Bayesian statistics. If \((Z_1, \ldots ,Z_N)\) is a finitely exchangeable sequence of N random variables taking values in some Polish space X, we show that the law \(\mu _k\) of the first k components has a representation of the form $$\begin{aligned} \mu _k=\int _{\mathcal{P}_{\frac{1}{N}}(X)} F_{N,k}(\lambda ) \, \text{ d } \alpha (\lambda ) \end{aligned}$$for some probability measure \(\alpha \) on the set of \(\frac{1}{N}\)-quantized probability measures on X and certain universal polynomials \(F_{N,k}\). The latter consist of a leading term \(N^{k-1}\! /\prod _{j=1}^{k-1}(N\! -\! j) \lambda ^{\otimes k}\) and a finite, exponentially decaying series of correlated corrections of order \(N^{-j}\) (\(j=1, \ldots ,k\)). The \(F_{N,k}(\lambda )\) are precisely the extremal such laws, expressed via an explicit polynomial formula in terms of their one-point marginals \(\lambda \). Applications include novel approximations of MMOT via polynomial convexification and the identification of the remainder which is estimated in the celebrated error bound of Diaconis and Freedman (Ann Probab 8(4):745–764, 1980) between finite and infinite exchangeable laws.

The semi-classical limit of large fermionic systems

We study a system of N fermions in the regime where the intensity of the interaction scales as 1 / N and with an effective semi-classical parameter $$\hbar =N^{-1/d}$$ where d is the space dimension. For a large class of interaction potentials and of external electromagnetic fields, we prove the convergence to the Thomas–Fermi minimizers in the limit $$N\rightarrow \infty $$ . The limit is expressed using many-particle coherent states and Wigner functions. The method of proof is based on a fermionic de Finetti–Hewitt–Savage theorem in phase space and on a careful analysis of the possible lack of compactness at infinity.

Savage's theorem under changing awareness

This paper proposes a simple unified framework of choice under changing awareness, addressing both outcome awareness and (nature) state awareness, and both how fine and how exhaustive the awareness is. Six axioms characterize an (essentially unique) expected-utility rationalization of preferences, in which utilities and probabilities are revised according to three revision rules when awareness changes: (R1) utilities of unaffected outcomes are transformed affinely; (R2) probabilities of unaffected events are transformed proportionally; (R3) enough probabilities ‘objectively’ never change (they represent revealed objective risk). Savage's Theorem is a special case of the theorem, namely the special case of fixed awareness, in which our axioms reduce to Savage's axioms while R1 and R2 hold trivially and R3 reduces to Savage's requirement of atomless probabilities. Rule R2 parallels Karni and Viero's (2013) ‘reverse Bayesianism’ and Ahn and Ergin's (2010) ‘partition-dependence’. The theorem draws mathematically on Kopylov (2007), Niiniluoto (1972) and Wakker (1981).

Risk-sensitive mean-field-type games with [formula omitted]-norm drifts

We study how risk-sensitive players act in situations where the outcome is influenced not only by the state–action profile but also by the distribution of it. In such interactive decision-making problems, the classical mean-field game framework does not apply. We depart from most of the mean-field games literature by presuming that a decision-maker may include its own-state distribution in its decision. This leads to the class of mean-field-type games. In mean-field-type situations, a single decision-maker may have a big impact on the mean-field terms for which new type of optimality equations are derived. We establish a finite dimensional stochastic maximum principle for mean-field-type games where the drift functions have a p-norm structure which weaken the classical Lipschitz and differentiability assumptions. Sufficient optimality equations are established via Dynamic Programming Principle but in infinite dimension. Using de Finetti–Hewitt–Savage theorem, we show that a propagation of chaos property with virtual particles holds for the non-linear McKean–Vlasov dynamics.

Asymptotic arbitrage with small transaction costs

We give characterizations of asymptotic arbitrage of the first and second kind and of strong asymptotic arbitrage for a sequence of financial markets with small proportional transaction costs λn on market n, in terms of contiguity properties of sequences of equivalent probability measures induced by λn-consistent price systems. These results are analogous to the frictionless case; compare (Kabanov and Kramkov in Finance Stoch. 2:143–172, 1998; Klein and Schachermayer in Theory Probab. Appl. 41:927–934, 1996). Our setting is simple, each market n contains two assets. The proofs use quantitative versions of the Halmos–Savage theorem (see Klein and Schachermayer in Ann. Probab. 24:867–881, 1996) and a monotone convergence result for nonnegative local martingales. Moreover, we study examples of models which admit a strong asymptotic arbitrage without transaction costs, but with transaction costs λn>0 on market n; there does not exist any form of asymptotic arbitrage. In one case, (λn) can even converge to 0, but not too fast.

The Microscopic Foundations of Vlasov Theory for Jellium-Like Newtonian $$N$$ N -Body Systems

The kinetic equations of Vlasov theory, in the weak formulation, are rigorously shown to govern the \(N\rightarrow \infty \) limit of the Newtonian dynamics of \(D\ge 2\)-dimensional \(N\)-body systems with attractive harmonic pair interactions and locally integrable repulsive inverse power law pair interactions, provided a mild higher moment hypothesis on the forces (which is shown to propagate globally in time for each \(N\)) will hold uniformly in \(N\) at later times if it holds uniformly in \(N\) initially (the uniformity in \(N\) of this moment condition is demonstrated to hold for an open set of initial data). Logarithmic interactions are included as a limiting case. The proof is based on the Liouville equation, more precisely the first member of the pertinent BBGKY hierarchy, and does not invoke the Hewitt–Savage theorem, nor any regularization of the interactions. In addition, a rigorous proof of the virial theorem and of some of its interesting ramifications is given.

Red herrings: Some thoughts on the meaning of zero-probability events and mathematical modeling

Kicking off the discussion following Savage's presentation at the 1952 Paris colloquium, Arrow raised what he considered to be a difficulty with the intuitive interpretation of Savage's theorem. It suggests that decision makers strictly prefer betting on an event of measure zero over betting on a proper subset of that event. Within the realm of the revealed-preference methodology and limited verifiability, Arrow's difficulty is a red herring: the problem he poses has its origin in the technical aspects of Savage's model and not in its substantive aspect.

A note on the Chernoff–Savage theorem

Abstract This note reconsiders the proof of the celebrated Chernoff–Savage (Ann. Math. Statist. 29, 1958) theorem as given in Puri and Sen (Nonparametric Methods in Multivariate Analysis, 1971). While most arguments in the proof presented there remain correct, some serious errors have occurred in the technical content dealing with the asymptotic negligibility of the C2N-component of the remainder (in their notation) terms. The purpose of the present note is to present a short corrected proof that C 2N =O P (N - 1 2 ) as N→∞.

A note on Savage's theorem with a finite number of states

This article gives a preference-based characterization of subjective expected utility for the general equilibrium model with a finite number of states. The characterization follows Savage (1954) as closely as possible but has to abandon his axiom (P6), atomlessness of events, since this requires an infinite state space. To introduce continuity we replace (P6) with a continuity assumption on the set of consequences and assume the preferences are smooth. Then we apply Savage's sure-thing principle and his state-independence axiom to get an additively separable utility representation. Finally, to separate subjective probabilities from basic tastes, we apply a new axiom, which states that for each pair of states the marginal rate of substitution is constant along the certainty line.

RANK TESTS IN 2X2 DESIGNS

Abstract. In literature numerous attempts can be found for the evaluation of two factor designs with fixed effects by means of rank tests. The aim of the present article is to show the limits of these methods and to give some new procedures for 2X2 designs. First, functionals of distribution functions shall be defined whose relations to the usual parameters of the linear model are analysed. These functionals are free of nuissance parameters under the respective hypothesis; they are estimated by special ranks of the data. The asymptotic distribution of these statistics is derived by a generalization of the Chemoff–Savage theorem for correlated random variables. The asymptotic variance depends on the parent distribution function but it can be estimated by using special rank methods. Thus, one obtains asymptotically distribution–free tests for two–factor designs with fixed effects. Some counter examples show why it is not possible to construct suitable rank tests for greater designs than the 2X2 design. The paper closes with a discussion of the drawbacks of the well known rank transform.

A Note on Fine and Tight Qualitative Probabilities

Savage (1954) has shown that fine and tight qualitative probabilities are realizable by finitely additive probability measures. His proof for this result is, however, in need of a correction. Fine qualitative probabilities are either atomless or equivalent to the union of $n$ equivalent atoms. Tight qualitative probabilities are always atomless. Qualitative probability structures, which are equivalent to the union of $n$ equivalent atoms, are realizable by a unique probability measure. Fine qualitative probabilities are almost realizable. With these results, the proof for Savage's theorem can be worked out and a theorem of Villegas (1964) can be strengthened.

Weak Convergence and a Chernoff-Savage Theorem for Random Sample Sizes

A Chernoff and Savage theorem on the asymptotic normality of 2-sample linear rank statistics is here established for random sample sizes. The proof parallels that of Pyke and Shorack (1968), hereafter referred to as PS. A mild restriction on the underlying distributions is needed in the present situation. A result of Pyke (1968) on the weak convergence of the 1-sample empirical process for random sample sizes in the ordinary uniform metric is here extended to other metrics. This extension provides an essential step in the present proof and is also of separate interest. The results extend immediately to $c$-samples.