Martingale Methods for Patterns and Scan Statistics

Pozdnyakov, Vladimir; Steele, J. Michael

doi:10.1007/978-0-8176-4749-0_14

Cited by 8 publications

(4 citation statements)

References 17 publications

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“…However, if one is only interested in τ (w,s) and not X (w,s) , then there exist other methods to compute E(τ (w,s) ), and also in principle the full distribution of τ (w,s) -see e.g. [23] or [24] which give formulae to obtain the probability generating function of τ (w,s) . To our knowledge, however, these also result in a large system of equations, and not the ending pattern distribution.…”

Section: B Related Workmentioning

confidence: 99%

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Tools for the Analysis of Quantum Protocols Requiring State Generation Within a Time Window

Davies,

Beauchamp,

Vardoyan

et al. 2024

IEEE Trans. Quantum Eng.

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Quantum protocols commonly require a certain number of quantum resource states to be available simultaneously. An important class of examples is quantum network protocols that require a certain number of entangled pairs. Here, we consider a setting in which a process generates a quantum resource state with some probability p in each time step, and stores it in a quantum memory that is subject to timedependent noise. To maintain sufficient quality for an application, each resource state is discarded from the memory after w time steps. Let s be the number of desired resource states required by a protocol. We characterise the probability distribution X (w,s) of the ages of the quantum resource states, once s states have been generated in a window w. Combined with a time-dependent noise model, knowledge of this distribution allows for the calculation of fidelity statistics of the s quantum resources. We also give exact solutions for the first and second moments of the waiting time τ (w,s) until s resources are produced within a window w, which provides information about the rate of the protocol. Since it is difficult to obtain general closed-form expressions for statistical quantities describing the expected waiting time E(τ (w,s) ) and the distribution X (w,s) , we present two novel results that aid their computation in certain parameter regimes. The methods presented in this work can be used to analyse and optimise the execution of quantum protocols. Specifically, with an example of a Blind Quantum Computing (BQC) protocol, we illustrate how they may be used to infer w and p to optimise the rate of successful protocol execution. INDEX TERMS Quantum networks, performance analysis, scan statisticsThis article has been accepted for publication in IEEE Transactions on Quantum Engineering.

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Section: B Related Workmentioning

confidence: 99%

“…Further, in Appendix B-4 we give a formula for the second moment of τ (w,s) , which now involves two linear systems of size |Ω(w, s)|. A martingale method is also used for its derivation, and for this we refer to [24]. The second moment of τ (w,s) can then be used to calculate the variance and standard deviation of τ (w,s) .…”

Section: A Infinite Windowmentioning

confidence: 99%

Tools for the Analysis of Quantum Protocols Requiring State Generation Within a Time Window

Davies,

Beauchamp,

Vardoyan

et al. 2024

IEEE Trans. Quantum Eng.

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“…For a survey of the area, we refer the interested reader to Balakrishnan and Koutras [1] and to the edited volume of Glaz et al [12]. In Pozdnyakov et al [17] and Pozdnyakov and Steele [16], a martingale approach (for other approaches, see, e.g. [9], [10], and [19]) has been exploited for scan and pattern related problems.…”

Section: Introductionmentioning

confidence: 99%

New maximal inequalities for N-demimartingales with scan statistic applications

Koutras

Lyberopoulos

2017

J. Appl. Probab.

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In the present work, some new maximal inequalities for nonnegative N-demi(super)martingales are first developed. As an application, new bounds for the cumulative distribution function of the waiting time for the first occurrence of a scan statistic in a sequence of independent and identically distributed (i.i.d.) binary trials are obtained. A numerical study is also carried out for investigating the behavior of the new bounds.

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“…A development of the gambling team method to many teams of gamblers was introduced by Pozdnyakov et al [8] (see also [9]). It can be used to compute higher moments, generating functions of the waiting time and to calculate probabilities for scan statistics (see [8,10]). In [5], [7] and [1] one can find an application of the method of gambling team to investigations of occurrences of patterns in Markov chains.…”

Section: Introduction and Notationmentioning

confidence: 99%

A note on the gambling team method

Zajkowski

2014

Statistics & Probability Letters

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Gerber and Li in [4] formulated, using a Markov chain embedding, a system of equations that describes relations between generating functions of waiting time distributions for occurrences of patterns in a sequence of independent repeated experiments when initial outcomes of the process are known. We show how this system of equations can be obtained by using the classical gambling team technique . We also present a form of solution of the system and give an example showing how first results of trials influence the probabilities that a chosen pattern precedes remaining ones in a realization of the process.2010 Mathematics Subject Classification: 60G40, 60G42.

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Martingale Methods for Patterns and Scan Statistics

Cited by 8 publications

References 17 publications

Tools for the Analysis of Quantum Protocols Requiring State Generation Within a Time Window

Tools for the Analysis of Quantum Protocols Requiring State Generation Within a Time Window

New maximal inequalities for N-demimartingales with scan statistic applications

A note on the gambling team method

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