Moments of infinite convolutions of symmetric Bernoulli distributions

Escribano, Carmen López; Sastre, María Asunción; Torrano, Emilio

doi:10.1016/s0377-0427(02)00595-2

Cited by 15 publications

(10 citation statements)

References 10 publications

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“…Having proven the first equality in (27), the first equality in (28) then follows from the bounded convergence theorem upon noting that Finally, the first equality in (29) follows from applying the same argument that gave the first equality in (27) to the integrals Proof of Theorem 2. Equations ( 30) and (33) follow immediately from the definition of C(t) in (57).…”

Section: Appendixmentioning

confidence: 98%

“…We note that the random variables A and B in (24) generalize a class of random variables known as infinite Bernoulli convolutions. In particular, in the special case that f is the single dose protocol and {ξ n } n∈Z are iid, then A and B are (after a linear transformation) standard infinite Bernoulli convolutions, which are known for their very irregular distributions and have been studied in the pure math literature for many decades [21][22][23][26][27][28][29][30].…”

Section: General Drug Level Statisticsmentioning

confidence: 99%

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Designing drug regimens that mitigate nonadherence

Counterman¹,

Lawley²

2021

Preprint

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Medication adherence is a well-known problem for pharmaceutical treatment of chronic diseases. Understanding how nonadherence affects treatment efficacy is made difficult by the ethics of clinical trials that force patients to skip medication doses, the unpredictable timing of missed doses by actual patients, and the many competing variables that can either mitigate or magnify the deleterious effects of nonadherence, such as pharmacokinetic absorption and elimination rates, dosing intervals, dose sizes, adherence rates, etc. In this paper, we formulate and analyze a mathematical model of the drug concentration in an imperfectly adherent patient. Our model takes the form of the standard single compartment pharmacokinetic model with first order absorption and elimination, except that the patient takes medication only at a given proportion of the prescribed dosing times. Doses are missed randomly, and we use stochastic analysis to study the resulting random drug level in the body. We then use our mathematical results to propose principles for designing drug regimens that are robust to nonadherence. In particular, we quantify the resiliency of extended release drugs to nonadherence, which is quite significant in some circumstances, and we show the benefit of taking a double dose following a missed dose if the drug absorption or elimination rate is slow compared to the dosing interval. We further use our results to compare some antiepileptic and antipsychotic drug regimens.

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Section: Appendixmentioning

confidence: 98%

Section: General Drug Level Statisticsmentioning

confidence: 99%

Designing drug regimens that mitigate nonadherence

Counterman¹,

Lawley²

2021

Preprint

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“…According to the Itô-type formulas [8, Theorems 1.5 and 1.10], the most interesting case is the one in which p = 1/H is an even integer. For this case, Theorem 1 from Escribano et al [11] provides an exact formula for E[|Z H | p ] in terms of Bernoulli numbers B 2k and partitions of n := p/2, namely,…”

Section: Introductionmentioning

confidence: 99%

On (signed) Takagi–Landsberg functions: pth variation, maximum, and modulus of continuity

Mishura

Schied

2019

Journal of Mathematical Analysis and Applications

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“…Our method leads to similar transformations for the associated Hessenberg matrices. Our work is also related to the problem of Bernoulli convolutions [6,13,19].…”

Section: Introductionmentioning

confidence: 99%

Computing the Hessenberg matrix associated with a self-similar measure

Escribano

Giraldo

Sastre

et al. 2011

Journal of Approximation Theory

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We introduce in this paper a method to calculate the Hessenberg matrix of a sum of measures from the Hessenberg matrices of the component measures. Our method extends the spectral techniques used by G. Mantica to calculate the Jacobi matrix associated with a sum of measures from the Jacobi matrices of each of the measures.We apply this method to approximate the Hessenberg matrix associated with a self-similar measure and compare it with the result obtained by a former method for self-similar measures which uses a fixed point theorem for moment matrices. Results are given for a series of classical examples of self-similar measures.Finally, we also apply the method introduced in this paper to some examples of sums of (not self-similar) measures obtaining the exact value of the sections of the Hessenberg matrix.

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Moments of infinite convolutions of symmetric Bernoulli distributions

Cited by 15 publications

References 10 publications

Designing drug regimens that mitigate nonadherence

Designing drug regimens that mitigate nonadherence

On (signed) Takagi–Landsberg functions: pth variation, maximum, and modulus of continuity

Computing the Hessenberg matrix associated with a self-similar measure

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