On Marcinkiewicz–Zygmund laws

Szewczak, Zbigniew S.

doi:10.1016/j.jmaa.2010.10.011

Cited by 9 publications

(5 citation statements)

References 20 publications

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“…As Z is an all-one matrix, the rank Z T Y is 1, and λ max Z T Y = tr{Z T Y} = L l=1 F f =1 y l,f , where y l,f is the (l, f )-th entry in Y. It follows from the Marcinkiewicz-Zygmund strong law of large numbers [24] that, asymptotically almost surely, λ max Z T Y = o((nF ) 1/p ) for any 1 ≤ p < 2. Since F ≤ L = Θ(n), we have g(n) n λ max Z T Y = o F 1/p g(n)…”

Section: Appendixmentioning

confidence: 99%

Network anti-inference: A fundamental perspective on proactive strategies to counter flow inference

Wang

2015

2015 IEEE Conference on Computer Communications (INFOCOM)

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Network inference is an effective mechanism to infer end-to-end flow rates and has enabled a variety of applications (e.g., network surveillance and diagnosis). The paper is focused on the opposite side of network inference, i.e., how to make inference inaccurate, which we call network anti-inference. As most research efforts have been focused on developing efficient inference methods, design of anti-inference is largely overlooked. Anti-inference scenarios can rise when network inference is not desirable, such as in clandestine communication and military applications. Our objective is to explore network dynamics to provide anti-inference. In particular, we consider two proactive strategies that cause network dynamics: transmitting deception traffic and changing routing to mislead the inference. We build an analytical framework to quantify the induced inference errors of the proactive strategies that maintain limited costs. We find via analysis and simulations that for deception traffic, a simple random transmission strategy can achieve inference errors on the same order of the best coordinated transmission strategy; while changing routing can cause inference errors of higher order than any deception traffic strategy. Our results not only reveal the fundamental perspective on proactive strategies, but also offer the guidance into practical design of anti-inference.

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Section: Appendixmentioning

confidence: 99%

Network anti-inference: A fundamental perspective on proactive strategies to counter flow inference

Wang

2015

2015 IEEE Conference on Computer Communications (INFOCOM)

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“…where the last step holds because ‖ ( )−1 − ( −1) ‖ ≤ /( +1) by the definition of ( ) and (26), (27), and the subadditivity of ‖ ⋅ ‖, we know that…”

Section: Corollarymentioning

confidence: 99%

“…As information, we mention that Baum-Katz type theorems in different dependent setups have been studied by many authors. For example, Li et al [24] studied moving average processes; Shao [25,26], Szewczak [27] considered mixing conditions; Baek and Park [28] studied negatively dependent random variables; Liang [29], Liang and Su [30], Kuczmaszewska [31], Kruglov [32], and Ko [33] studied negatively associated random variables.…”

Section: Introductionmentioning

confidence: 99%

Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences

Hao

2013

Abstract and Applied Analysis

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We study the convergence rates in the law of large numbers for arrays of Banach valued martingale differences. Under a simple moment condition, we show sufficient conditions about the complete convergence for arrays of Banach valued martingale differences; we also give a criterion about the convergence for arrays of Banach valued martingale differences. In the special case where the array of Banach valued martingale differences is the sequence of independent and identically distributed real valued random variables, our result contains the theorems of Hsu-Robbins-Erdös (1947, 1949, and 1950), Spitzer (1956), and Baum and Katz (1965). In the real valued single martingale case, it generalizes the results of Alsmeyer (1990). The consideration of Banach valued martingale arrays (rather than a Banach valued single martingale) makes the results very adapted in the study of weighted sums of identically distributed Banach valued random variables, for which we prove new theorems about the rates of convergence in the law of large numbers. The results are established in a more general setting for sums of infinite many Banach valued martingale differences. The obtained results improve and extend those of Ghosal and Chandra (1998).

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“…Weak laws of large numbers with the norming constants are of the form n 1/αL (n 1/α ) were studied by Gut [21], and Matsumoto and Nakata [33]. The Marcinkiewicz-Zygmund strong law of large numbers has been extended and generalized in many directions by a number of authors, see [1,12,22,24,38,39,46] and references therein. To our best knowledge, there is not any result in the literature that considers strong law of large numbers with general normalizing constants n 1/αL (n 1/α ) except Gut and Stadmüller [22] who studied the Kolmogorov strong law of large numbers, but for delay sums.…”

Section: Introductionmentioning

confidence: 99%

The Marcinkiewicz–Zygmund-Type Strong Law of Large Numbers with General Normalizing Sequences

Anh¹,

Hiến

Thành

et al. 2019

J Theor Probab

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This paper establishes complete convergence for weighted sums and the Marcinkiewicz-Zygmundtype strong law of large numbers for sequences of negatively associated and identically distributed random variables {X, X n , n ≥ 1} with general normalizing constants under a moment condition that ER(X) < ∞, where R(•) is a regularly varying function. The result is new even when the random variables are independent and identically distributed (i.i.d.), and a special case of this result comes close to a solution to an open question raised by Chen and Sung (Statist Probab Lett 92:45-52, 2014). The proof exploits some properties of slowly varying functions and the de Bruijin conjugates. A counterpart of the main result obtained by Martikainen (J Math Sci 75(5):1944-1946 on the Marcinkiewicz-Zygmund-type strong law of large numbers for pairwise i.i.d. random variables is also presented. Two illustrated examples are provided, including a strong law of large numbers for pairwise negatively dependent random variables which have the same distribution as the random variable appearing in the St. Petersburg game.

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On Marcinkiewicz–Zygmund laws

Abstract: Continued fractionsMarcinkiewicz-Zygmund laws with convergence rates are established here for a class of strictly stationary ergodic sequences.

Cited by 9 publications

References 20 publications

Network anti-inference: A fundamental perspective on proactive strategies to counter flow inference

Network anti-inference: A fundamental perspective on proactive strategies to counter flow inference

Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences

The Marcinkiewicz–Zygmund-Type Strong Law of Large Numbers with General Normalizing Sequences

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