An Inequality and Almost Sure Convergence

Kounias, Eustratios G.; Weng, Teng-Shan

doi:10.1214/aoms/1177697615

Cited by 27 publications

(18 citation statements)

References 1 publication

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“…I n what follows Lemma 2 has been found by KOUNIAS and WENG [5]. I n what follows Lemma 2 has been found by KOUNIAS and WENG [5].…”

Section: B -1supporting

confidence: 65%

Some Inequalities for the Maximum of Partial Sums of Random Variables

Mogyoródi¹

1975

Mathematische Nachrichten

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1. In this paper we shall give some moment and probability inequalities for the supremum of the arithmetic mean of random variables. These are in some special cases arbitrary but, in general, we shall restrict ourselves to the case of martingales or of totally independent random variables. By the aid of these inequalities we deduce some strong laws of large numbers.In the sequel we shall use the following results. Let So = 0, Si, S,, . . . be a martingale. In his paper [ 13 BURKHOLDER proved the following inequality : if p > 1 then there exist constants Cp and cp such that holds, where x k = X ks k -i , Ic = 1 , 2 , . . . is the martingale difference sequence. As a consequence, we obtain for p > 1 the famous inequality of MARCIN- KIEWICZ and ZYGMUND for totally independent random variables X , with E(X,) = 0, k = 1, 2, . . . In the latter case it is known that (1) holds also forFor 1 5 p 5 2 we obtain by the well-known C,-inequality that P n ~((~8x:)i) 2 i= c 1 E(lxilp) * Also, for p > 2 using the inequality of HOLDER we get Thus, if So = 0, S,, S,, . . . is a martingale, then for p > 1 we obtain by (1)with a constant Cp and with a = max 0, ~--1 . c 1 If p 5 1 and Xi, X,, . . . is an arbitrary sequence of random variables, then, as it is known, ( 2 ) holds by the C,-inequality.

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“…I n what follows Lemma 2 has been found by KOUNIAS and WENG [5]. I n what follows Lemma 2 has been found by KOUNIAS and WENG [5].…”

Section: B -1supporting

confidence: 65%

Some Inequalities for the Maximum of Partial Sums of Random Variables

Mogyoródi¹

1975

Mathematische Nachrichten

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“…Remark 1. These conditions on (κ n ) n∈N have been deduced from conditions for strong law of large numbers obtained in [34] and are not too restrictive: for instance, if the Lipschitz coefficients of f θ , M θ (the case using H θ can be treated similarly) and their derivatives are bounded by a geometric or Riemanian decrease:…”

Section: Assumption K(θ)mentioning

confidence: 99%

“…which is finite by assumption K(Θ), and this achieves the proof. 2/ If X ⊂ AC( H θ ) and using Corollary 1 of [34], with r ≤ 4, (7.1) is established when:…”

Section: Consistent Model Selection Criteria and Goodness-of-fit Test For Common Time Series2035mentioning

confidence: 99%

Consistent model selection criteria and goodness-of-fit test for common time series models

Bardet¹,

Kamila²,

Kengne³

2020

Electron. J. Statist.

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This paper studies the model selection problem in a large class of causal time series models, which includes both the ARMA or AR(∞) processes, as well as the GARCH or ARCH(∞), APARCH, ARMA-GARCH and many others processes. To tackle this issue, we consider a penalized contrast based on the quasi-likelihood of the model. We provide sufficient conditions for the penalty term to ensure the consistency of the proposed procedure as well as the consistency and the asymptotic normality of the quasi-maximum likelihood estimator of the chosen model. We also propose a tool for diagnosing the goodness-of-fit of the chosen model based on a Portmanteau test. Monte-Carlo experiments and numerical applications on illustrative examples are performed to highlight the obtained asymptotic results. Moreover, using a data-driven choice of the penalty, they show the practical efficiency of this new model selection procedure and Portemanteau test.

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“…Kounias and Weng [17] proved the following Hájek-Rényi type inequality for arbitrary random variables (i.e. without assuming any dependence condition).…”

Section: Some Classical Hájek-rényi Type Maximal Inequalitiesmentioning

confidence: 99%