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In, Nasiraee et al. showed that, despite previous assumptions, when the integrand is an arbitrary function, Carleman's condition is not sufficient, as demonstrated by a counter-example. In fact, the example violates the bijection, i.e. determinacy, property in the probability sum theorem. When the
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M. Nasiraee, Jav. Kazemitabar and Jal. Kazemitabar, "The
Bijection Property in the Law of Total Probability and Its Application in Communication Theory," in IEEE Communications Letters, doi: 10.1109/LCOMM.2024.3447352.
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S. S. Shamai, “Capacity of a pulse amplitude modulated direct detection photon channel,” IEE Proceedings I (Communications, Speech and Vision), vol. 137, no. 6, pp. 424–430, Dec. 1990.
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integrand is an arbitrary function, they further establish a sufficient condition for the determinacy of the moment problem, referred to as the
638:. 5th ed. Cambridge Series in Statistical and Probabilistic Mathematics 49. Cambridge ; New York, NY: Cambridge University Press, 2019.
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267:{\displaystyle m_{n}=\int _{-\infty }^{+\infty }x^{n}\,d\mu (x)~,\quad n=0,1,2,\cdots }
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114:(the moment problem on the whole real line), the theorem states the following:
350:{\displaystyle \sum _{n=1}^{\infty }m_{2n}^{-{\frac {1}{2n}}}=+\infty ,}
554:{\displaystyle \sum _{n=1}^{\infty }m_{n}^{-{\frac {1}{2n}}}=+\infty .}
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The
Classical Moment Problem and Some Related Questions in Analysis
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satisfies
Carleman's condition, there is no other measure
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gives a sufficient condition for the determinacy of the
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574:generalized Carleman's condition
567:Generalized Carleman's condition
98:The condition was discovered by
634:Chapter 3.3, Durrett, Richard.
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469:as its sequence of moments.
429:{\displaystyle \mathbb {R} }
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626:Akhiezer, N. I. (1965).
479:Stieltjes moment problem
473:Stieltjes moment problem
112:Hamburger moment problem
106:Hamburger moment problem
462:{\displaystyle (m_{n})}
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91:{\displaystyle \mu .}
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657:Moment (mathematics)
630:. Oliver & Boyd.
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394:; that is,
392:determinate
646:Categories
620:References
546:∞
520:−
505:∞
490:∑
402:μ
342:∞
316:−
298:∞
283:∑
262:⋯
219:μ
200:∞
192:∞
189:−
185:∫
125:μ
102:in 1922.
83:μ
59:ν
39:μ
477:For the
110:For the
18:analysis
139:measure
73:moments
30:measure
231:
580:Notes
436:with
137:be a
117:Let
390:is
141:on
75:as
648::
576:.
20:,
549:.
543:+
540:=
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529:2
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256:2
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250:1
247:,
244:0
241:=
238:n
234:,
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225:x
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216:d
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206:x
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150:R
86:.
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