119,008 research outputs found

    The Sharp Lower Bound of Asymptotic Efficiency of Estimators in the Zone of Moderate Deviation Probabilities

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    For the zone of moderate deviation probabilities the local asymptotic minimax lower bound of asymptotic efficiency of estimators is established. The estimation parameter is multidimensional. The lower bound admits the interpretation as the lower bound of asymptotic efficiency in confidence estimation

    The Guarded Lambda-Calculus: Programming and Reasoning with Guarded Recursion for Coinductive Types

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    We present the guarded lambda-calculus, an extension of the simply typed lambda-calculus with guarded recursive and coinductive types. The use of guarded recursive types ensures the productivity of well-typed programs. Guarded recursive types may be transformed into coinductive types by a type-former inspired by modal logic and Atkey-McBride clock quantification, allowing the typing of acausal functions. We give a call-by-name operational semantics for the calculus, and define adequate denotational semantics in the topos of trees. The adequacy proof entails that the evaluation of a program always terminates. We introduce a program logic with L\"ob induction for reasoning about the contextual equivalence of programs. We demonstrate the expressiveness of the calculus by showing the definability of solutions to Rutten's behavioural differential equations.Comment: Accepted to Logical Methods in Computer Science special issue on the 18th International Conference on Foundations of Software Science and Computation Structures (FoSSaCS 2015

    Moment conditions in strong laws of large numbers for multiple sums and random measures

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    The validity of the strong law of large numbers for multiple sums SnS_n of independent identically distributed random variables ZkZ_k, knk\leq n, with rr-dimensional indices is equivalent to the integrability of Z(log+Z)r1|Z|(\log^+|Z|)^{r-1}, where ZZ is the typical summand. We consider the strong law of large numbers for more general normalisations, without assuming that the summands ZkZ_k are identically distributed, and prove a multiple sum generalisation of the Brunk--Prohorov strong law of large numbers. In the case of identical finite moments of irder 2q2q with integer q1q\geq1, we show that the strong law of large numbers holds with the normalisation n1nr1/2(logn1lognr)1/(2q)+ε\|n_1\cdots n_r\|^{1/2}(\log n_1\cdots\log n_r)^{1/(2q)+\varepsilon} for any ε>0\varepsilon>0. The obtained results are also formulated in the setting of ergodic theorems for random measures, in particular those generated by marked point processes.Comment: 15 page
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