multilevel large deviations and interacting diffusions let n be a sequence of random variables with values in a topological space which satisfy the large deviation principle for eachm and eachn let m n denote the empirical measure associated withm independent copies of n as a main result we show that m n also satisfies the large deviation principle asm n we derive several representations of the associated rate function these results are then applied to empirical measure processes m n t m 1 i 1n in t 0 t t where 1n mn t is a system of weakly interacting diffusions with noise intensity 1 n this is a continuation of our previous work on the mckean vlasov limit and related hierarchical models 4 5

evolutionary games in economics evolutionary games are introduced as models for repeated anonymous strategic interaction actions or behaviors which are more fit given the current distribution of behaviors tend over time to displace less fit behaviors cone fields characterize the continuous time processes compatible with a given fitness or payoff function for large classes of dynamics it is shown that all stable steady states are nash equilibria and that all nash equilibria are steady states the biologists evolutionarily stable strategy condition is shown to be less closely related to the dynamic equilibria economic examples and a literature survey are also provided copyright 1991 by the econometric society

some asymptotic theory for the bootstrap efron s bootstrap method of distribution approximation is shown to be asymptotically valid in a large number of situations including t statistics the empirical and quantile processes and von mises functionals some counter examples are also given to show that the approximation does not always succeed

weyl solutions and j selfadjointness for dirac operators we consider a non selfadjoint dirac type differential expression 0 1 d q y j n dy over dx q x y with a non selfadjoint potential matrix q epsilon l 1 loc scr j bbb c n times n and a signature matrix j n j j n j n epsilon bbb c n times n here scr j denotes either the line bbb r or the half line bbb r with this differential expression one associates in l 2 scr j bbb c n the closed maximal and minimal operators d max q and d min q respectively one of our main results for the whole line case states that d max q d min q in l 2 bbb r bbb c n moreover we show that if the minimal operator d min q in l 2 bbb r bbb c n is j symmetric with respect to an appropriate involution j then it is j selfadjoint similar results are valid in the case of the semiaxis bbb r in particular we show that if n 2p and the minimal operator d min q in l 2 bbb r bbb c 2p is j symmetric then there exists a 2p times p weyl type matrix solution psi z cdot epsilon l 2 bbb r bbb c 2p times p of the equation d max q psi z cdot z psi z cdot a similar result is valid for the expression 0 1 whenever there exists a proper extension tilde a with dim dom tilde a dom d min q p and nonempty resolvent set in particular it holds if a potential matrix q has a bounded imaginary part this leads to the existence of a unique weyl function for the express 0 1 the main results are proven by means of a reduction to the self adjoint case by using the technique of dual pairs of operators the differential expression 0 1 is of significance as it appears in the lax formulation of the vector valued nonlinear schrodinger equation

j self adjoint extensions of a class of hamiltonian differential systems abstract this paper is concerned with j self adjoint extensions for a class of hamiltonian differential systems the domains of the corresponding minimal and maximal operators are described a classification of limit types of the systems at each of endpoints is given a property of the regularity field of the corresponding minimal operator is derived and an expression of the j defect index of the minimal operator by those of the left and right minimal operators is presented based on them a characterization of all j self adjoint extensions for the hamiltonian system is obtained in terms of square integrable solutions of a certain equation which is generated by this hamiltonian system and its formal adjoint as a consequence characterizations of all j self adjoint extensions are given for the systems in some special cases in particular all j self adjoint extensions are characterized in terms of square integrable solutions of the system itself in certain special cases

norm estimates of complex symmetric operators applied to quantum systems this paper communicates recent results in the theory of complex symmetric operators and shows through two non trivial examples their potential usefulness in the study of schrodinger operators in particular we propose a formula for computing the norm of a compact complex symmetric operator this observation is applied to two concrete problems related to quantum mechanical systems first we give sharp estimates on the exponential decay of the resolvent and the single particle density matrix for schrodinger operators with spectral gaps second we provide new ways of evaluating the resolvent norm for schrodinger operators appearing in the complex scaling theory of resonances

j self adjoint operators with mathcal c symmetries an extension theory approach a well known tool in conventional von neumann quantum mechanics is the self adjoint extension technique for symmetric operators it is used e g for the construction of dirac hermitian hamiltonians with point interaction potentials here we reshape this technique to allow for the construction of pseudo hermitian j self adjoint hamiltonians with complex point interactions we demonstrate that the resulting hamiltonians are bijectively related to the so called hypermaximal neutral subspaces of the defect krein space of the symmetric operator this symmetric operator is allowed to have arbitrary but equal deficiency indices n n general properties of the operators for these hamiltonians are derived a detailed study of operator parametrizations and krein type resolvent formulae is provided for j self adjoint extensions of symmetric operators with deficiency indices 2 2 the technique is exemplified on 1d pseudo hermitian schr dinger and dirac hamiltonians with complex point interaction potentials

integration of large deviation kernels and applications to large deviations for evolutionary games we investigate large deviation properties for systems of probability kernels and these kernels when integrated with respect to a system of measures that also satisfy the large deviation principle the results of our analysis are then used to demonstrate the large deviation property and derive rate functions for systems of measures derived from the distributions of evolutionary game processes

statistical dynamics of the royal road genetic algorithm abstract metastability is a common phenomenon many evolutionary processes both natural and artificial alternate between periods of stasis and brief periods of rapid change in their behavior in this paper an analytical model for the dynamics of a mutation only genetic algorithm ga is introduced that identifies a new and general mechanism causing metastability in evolutionary dynamics the ga s population dynamics is described in terms of flows in the space of fitness distributions the trajectories through fitness distribution space are derived in closed form in the limit of infinite populations we then show how finite populations induce metastability even in regions where fitness does not exhibit a local optimum in particular the model predicts the occurrence of fitness epochs periods of stasis in population fitness distributions at finite population size and identifies the locations of these fitness epochs with the flow s hyperbolic fixed points this enables exact predictions of the metastable fitness distributions during the fitness epochs as well as giving insight into the nature of the periods of stasis and the innovations between them all these results are obtained as closed form expressions in terms of the ga s parameters an analysis of the jacobian matrices in the neighborhood of an epoch s metastable fitness distribution allows for the calculation of its stable and unstable manifold dimensions and so reveals the state space s topological structure more general quantitative features of the dynamics fitness fluctuation amplitudes epoch stability and speed of the innovations are also determined from the jacobian eigenvalues the analysis shows how quantitative predictions for a range of dynamical behaviors that are specific to the finite population dynamics can be derived from the solution of the infinite population dynamics the theoretical predictions are shown to agree very well with statistics from ga simulations we also discuss the connections of our results with those from population genetics and molecular evolution theory

the evolution of conventions the author shows how a group of individuals can learn to play a coordination game without any common knowledge and with only a small amount of rationality the game is repeated many times by different players each player chooses an optimal reply based on incomplete information about what other players have done in the past occasionally they make mistakes when the likelihood of mistakes is very small typically one coordination equilibrium will be played almost all of the time over the long run this stochastically stable equilibrium can be computed analytically using a general theorem the author proves on perturbed markov processes copyright 1993 by the econometric society

weighted variation inequalities for singular integrals and commutators abstract in this paper we extend the mixed weak type inequalities of sawyer type for calderon zygmund operators to the variation operators of type calderon zygmund operators moreover the corresponding quantitative weighted bounds as well as the weighted estimates in the extreme case p are also obtained meanwhile we also present the quantitative bloom type estimate and c p estimates for variation operators of commutators

learning mutation and long run equilibria in games an evolutionary model with a finite number of players and with stochastic mutations is analyzed the expansion and contraction of strategies is linked to their current relative success but mutuation perturbing the system from its deterministic evolution are present as well the focus is on the long run implications of ongoing mutations which drastically reduce the set of equilibria for 2 by 2 symmetric games with two symmetric strict nash equilibria the risk dominant equilibrium is selected in particular if both strategies have equal security levels the pareto dominant nash equilibrium is selected in particular if both strategies have equal security levels the pareto dominant nash equilibrium is selected even though there is another strict nash equilibrium copyright 1993 by the econometric society

on the strong limit point and dirichlet properties of second order differential expressions second order differential expressions of the form w 1 pf qf are considered at a singular end point some of the known relationships between properties such as dirichlet weak dirichlet and strong limit point are extended to incorporate an arbitrary positive weight function and complexvalued coefficients

on the boundary conditions characterizing j selfadjoint extensions of j symmetric operators the study of boundary value problems involving linear differentiai equations with real valued coefficients is by now a well established area of analysis on the other hand much less is known about the solvability of such problems when the coefficients or boundary conditions are known to be complex examples of this latter type arise naturally for example in nuclear physics e g the so called optical mode1 for low energy scattering 1 p ijo electromagnetic field theory dielectric waveguides with heat loss c f 19 i or the propagation of radio waves through inhomogeneous media s and elsewhere in cases like these the relevant differential expressions are no longer formally symmetric and hence the powerful methods associated with the spectral theory of selfadjoint operators are not available to facilitate the study of such problems glazman introduced in 4 the concept of a j symmetric operator in a complex hilbert space r let j be a given conjugation operator on 2 i e j is a conjugate linear involution with jx 3 u x for all x and y in 2 a closed densely defined linear operator t in 3 is said to be j symmetric if

oscillation and variation for singular integrals in higher dimensions in this paper we continue our investigations of square function inequalities in harmonic analysis here we investigate oscillation and variation inequalities for singular integral operators in dimensions d 1 our estimates give quantitative information on the speed of convergence of truncations of a singular integral operator including upcrossing and a jump inequalities

complex symmetric differential operators on fock space abstract the space of entire functions which are integrable with respect to the gaussian weight known also as the fock space is one of the preferred functional hilbert spaces for modeling and experimenting harmonic analysis quantum mechanics or spectral analysis phenomena this space of entire functions carries a three parameter family of canonical isometric involutions we characterize the linear differential operators acting on fock space which are complex symmetric with respect to these conjugations in parallel as a basis of comparison we discuss the structure of self adjoint linear differential operators the computation of the point spectrum of some of these operators is carried out in detail

the dynamics of a genetic algorithm for simple random ising systems abstract a formalism is presented for analysing genetic algorithms it is used to study a simple genetic algorithm consisting of selection mutation and crossover which is searching for the ground states of simple random ising spin systems a random field ideal paramagnet and a spin glass chain the formalism can also be applied to other population based search techniques and to biological models of micro evolution to make the problem tractable it is assumed that the population dynamics can be described by a few macroscopic order parameters and that the remaining microscopic degrees of freedom can be averaged out the macroscopic quantities that are used are the cumulants of the distribution of fitnesses or energies in the population a statistical mechanics model is presented which describes the population configuration in terms of the cumulants this is used to derive equations of motion for the cumulants predictions of the theory are compared with experiments and are shown to predict the average time to convergence and the average fitness of the final population accurately a simplified version of the equations is produced by keeping only leading nonlinear terms and truncating the cumulant expansion this is shown to give a novel description of the role of genetic operators in search e g it is argued that an important role of crossover is to reduce the skewness of the population

complex symmetric operators and applications ii a bounded linear operator t on a complex hilbert space h is called complex symmetric if t ct c where c is a conjugation an isometric antilinear involution of h we prove that t cj t where j is an auxiliary conjugation commuting with t t t we consider numerous examples including the poincare neumann singular integral bounded operator and the jordan model operator compressed shift the decomposition t cj t also extends to the class of unbounded c selfadjoint operators originally introduced by glazman in this context it provides a method for estimating the norms of the resolvents of certain unbounded operators

j self adjointness of a class of dirac type operators in this note we prove that the maximally defined operator associated with the dirac type differential expression m q iddxim q q ddxim where q represents a symmetric m m matrix i e q x q x a e with entries in l1loc r is j self adjoint where j is the antilinear conjugation defined by j 1c 1 0imim0 and c a1 am b1 bm a1 am b1 bm the differential expression m q is of significance as it appears in the lax formulation of the non abelian matrix valued focusing nonlinear schrodinger hierarchy of evolution equations

theoretical comparison of bootstrap confidence intervals we develop a unified framework within which many commonly used bootstrap critical points and confidence intervals may be discussed and compared in all seven different bootstrap methods are examined each being usable in both parametric and nonparametric contexts emphasis is on the way in which the methods cope with first and second order departures from normality percentile t and accelerated bias correction emerge as the most promising of existing techniques certain other methods are shown to lead to serious errors in coverage and position of critical point an alternative approach based on shortest bootstrap confidence intervals is developed we also make several more technical contributions in particular we confirm efron s conjecture that accelerated bias correction is second order correct in a variety of multivariate circumstances and give a simple interpretation of the acceleration constant

classification of sturm liouville differential equations with complex coefficients and operator realizations in this paper a new classification of sturm liouville differential equations with complex coefficients is given compared with the corresponding result of brown et al this classification reveals the great effects of rotation angle and it is independent of the rotation angles moreover the asymptotic behaviours of functions in the maximal domain are presented and j self adjoint extensions associated with the differential equations are characterized

large deviations of estimators the performance of a sequence of estimators t n of g theta can be measured by its inaccuracy rate lim inf n rightarrow infty n 1 log mathbb p theta t n g theta varepsilon for fixed varepsilon 0 optimality of consistent estimators operatorname wrt the inaccuracy rate is investigated it is shown that for exponential families in standard representation with a convex parameter space the maximum likelihood estimator is optimal if the parameter space is not convex which occurs for instance in curved exponential families in general no optimal estimator exists for the location problem the inaccuracy rate of m estimators is established if the underlying density is sufficiently smooth an optimal m estimator is obtained within the class of translation equivariant estimators tail behaviour of location estimators is studied a connection is made between gross error and inaccuracy rate optimality

on j selfadjoint extensions of j symmetric ordinary differential operators on donne une caracterisation des conditions aux limites qui determinent le domaine d une extension j autoadjointe d un operateur differentiel j symetrique t 0 qui a une extremite singuliere