The strong law of large numbers for sequential decisions under uncertainty

Algoet, P.

doi:10.1109/18.335876

Cited by 65 publications

(116 citation statements)

References 71 publications

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“…Accordingly, optimal prediction is formulated as the minimisation of the expected value of the predictor loss function [20,23,27,28,34]. For example, if {X t } is an arbitrary parametrised random source, the action a t =x t is set as next observation prediction, and the loss function is the squared distance l(a t , x t ) = E(a t − x t ) 2 then the optimal predictor will always choose the conditional mean as it is predicted value.…”

Section: Probabilistic Settingsmentioning

confidence: 99%

Statistical spectrum occupancy prediction for dynamic spectrum access: a classification

Eltom

Kandeepan

Evans

et al. 2018

J Wireless Com Network

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Spectrum scarcity due to inefficient utilisation has ignited a plethora of dynamic spectrum access solutions to accommodate the expanding demand for future wireless networks. Dynamic spectrum access systems allow secondary users to utilise spectrum bands owned by primary users if the resulting interference is kept below a pre-designated threshold. Primary and secondary user spectrum occupancy patterns determine if minimum interference and seamless communications can be guaranteed. Thus, spectrum occupancy prediction is a key component of an optimised dynamic spectrum access system. Spectrum occupancy prediction recently received significant attention in the wireless communications literature. Nevertheless, a single consolidated literature source on statistical spectrum occupancy prediction is not yet available in the open literature. Our main contribution in this paper is to provide a statistical prediction classification framework to categorise and assess current spectrum occupancy models. An overview of statistical sequential prediction is presented first. This statistical background is used to analyse current techniques for spectrum occupancy prediction. This review also extends spectrum occupancy prediction to include cooperative prediction. Finally, theoretical and implementation challenges are discussed.

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Section: Probabilistic Settingsmentioning

confidence: 99%

Statistical spectrum occupancy prediction for dynamic spectrum access: a classification

Eltom

Kandeepan

Evans

et al. 2018

J Wireless Com Network

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“…Bailey [5] (cf. Algoet [2] also) showed that any scheme that solves Problem 1 can be easily modified to solve Problem 4 (indeed, just exchange the data segment (X −n , . .…”

Section: Problemmentioning

confidence: 99%

Forward estimation for ergodic time series

Morvai

Weiss

2005

Annales de l'Institut Henri Poincare (B) Probability and Statis

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The forward estimation problem for stationary and ergodic time series {X n } ∞ n=0 taking values from a finite alphabet X is to estimate the probability that X n+1 = x based on the observations X i , 0 ≤ i ≤ n without prior knowledge of the distribution of the process {X n }. We present a simple procedure g n which is evaluated on the data segment (X 0 , . . . , X n ) and for which, error(n) = |g n (x) − P (X n+1 = x|X 0 , . . . , X n )| → 0 almost surely for a subclass of all stationary and ergodic time series, while for the full class the Cesaro average of the error tends to zero almost surely and moreover, the error tends to zero in probability.Le problème d'estimation future d'une série de temps ergodique et stationnaire {X n } ∞ n=0 , qui prend ses valeures dans un alphabet fini X , est d'estimer la probabilité que X n+1 = x, connaissant les X i pour 0 ≤ i ≤ n mais sans connaissance préalable de la distribution du processus {X i }. Nous présentons un procédé simple g n , evalué dur les données (X 0 , . . . , X n ), pour lequel erreur(n) = |g n (x) − P (X n+1 = x|X 0 , . . . , X n ) → 0 presque sûrement pour une sous-classe de toutes les séries de temps ergodiques et stationnaires, tandis que pour la classe entière la moyenne de Cesaro de l'erreur tend vers zéro presque sûrement. De plus, l'erreur tend vers zéro en probabilité.

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“…The fundamental limits, first published in [3], [1,2] reveal that the so-called log-optimal portfolio B * = {b * (·)} is the best possible choice for the maximization of S n . More precisely, on trading period n let b * (·) be such that…”

Section: The Log-optimal Portfolio Selectionmentioning

confidence: 99%

“…Note that the Table 6.1 contains a row with parameter λ = 0.5 which is not covered in the Theorem 5.1, however kernel-based Markowitz-type strategy can be used in this case too. The k and ℓ parameters of the best performing experts given in columns 2 − 5 are different: (2,10), (3,10), (2,8) and (1,1) respectively. The best performance of pairs of stocks was attained at different values of parameter λ (underlined in Table 6.1).…”

Section: Simulationmentioning

confidence: 99%

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An asymptotic analysis of the mean-variance portfolio selection

Ottucsák¹,

Vajda²

2007

Statistics &Amp; Decisions

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Summary: This paper gives an asymptotic analysis of the mean-variance (Markowitz-type) portfolio selection under mild assumptions on the market behavior. Theoretical results show the rate of underperformance of the risk aware Markowitz-type portfolio strategy in growth rate compared to the log-optimal portfolio strategy, which does not have explicit risk control. Statements are given with and without full knowledge of the statistical properties of the underlying process generating the market, under the only assumption that the market is stationary and ergodic. The experiments show how the achieved wealth depends on the coefficient of absolute risk aversion measured on past NYSE data.

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The strong law of large numbers for sequential decisions under uncertainty

Cited by 65 publications

References 71 publications

Statistical spectrum occupancy prediction for dynamic spectrum access: a classification

Statistical spectrum occupancy prediction for dynamic spectrum access: a classification

Forward estimation for ergodic time series

An asymptotic analysis of the mean-variance portfolio selection

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