scholarly journals Generalized Coupon Collection: The Superlinear Case

2011 ◽  
Vol 48 (01) ◽  
pp. 189-199 ◽  
Author(s):  
R. T. Smythe

We consider a generalized form of the coupon collection problem in which a random number,S, of balls is drawn at each stage from an urn initially containingnwhite balls (coupons). Each white ball drawn is colored red and returned to the urn; red balls drawn are simply returned to the urn. The question considered is then: how many white balls (uncollected coupons) remain in the urn after thekndraws? Our analysis is asymptotic asn→ ∞. We concentrate on the case whenkndraws are made, wherekn/n→ ∞ (the superlinear case), although we sketch known results for other ranges ofkn. A Gaussian limit is obtained via a martingale representation for the lower superlinear range, and a Poisson limit is derived for the upper boundary of this range via the Chen-Stein approximation.

2011 ◽  
Vol 48 (1) ◽  
pp. 189-199 ◽  
Author(s):  
R. T. Smythe

We consider a generalized form of the coupon collection problem in which a random number, S, of balls is drawn at each stage from an urn initially containing n white balls (coupons). Each white ball drawn is colored red and returned to the urn; red balls drawn are simply returned to the urn. The question considered is then: how many white balls (uncollected coupons) remain in the urn after the kn draws? Our analysis is asymptotic as n → ∞. We concentrate on the case when kn draws are made, where kn / n → ∞ (the superlinear case), although we sketch known results for other ranges of kn. A Gaussian limit is obtained via a martingale representation for the lower superlinear range, and a Poisson limit is derived for the upper boundary of this range via the Chen-Stein approximation.


2010 ◽  
Vol 42 (04) ◽  
pp. 994-1012 ◽  
Author(s):  
Hosam M. Mahmoud

In this paper we consider a generalized coupon collection problem in which a customer repeatedly buys a random number of distinct coupons in order to gather a large number n of available coupons. We address the following question: How many different coupons are collected after k = k n draws, as n → ∞? We identify three phases of k n : the sublinear, the linear, and the superlinear. In the growing sublinear phase we see o(n) different coupons, and, with true randomness in the number of purchases, under the appropriate centering and scaling, a Gaussian distribution is obtained across the entire phase. However, if the number of purchases is fixed, a degeneracy arises and normality holds only at the higher end of this phase. If the number of purchases have a fixed range, the small number of different coupons collected in the sublinear phase is upgraded to a number in need of centering and scaling to become normally distributed in the linear phase with a different normal distribution of the type that appears in the usual central limit theorems. The Gaussian results are obtained via martingale theory. We say a few words in passing about the high probability of collecting nearly all the coupons in the superlinear phase. It is our aim to present the results in a way that explores the critical transition at the ‘seam line’ between different Gaussian phases, and between these phases and other nonnormal phases.


2010 ◽  
Vol 42 (4) ◽  
pp. 994-1012 ◽  
Author(s):  
Hosam M. Mahmoud

In this paper we consider a generalized coupon collection problem in which a customer repeatedly buys a random number of distinct coupons in order to gather a large number n of available coupons. We address the following question: How many different coupons are collected after k = kn draws, as n → ∞? We identify three phases of kn: the sublinear, the linear, and the superlinear. In the growing sublinear phase we see o(n) different coupons, and, with true randomness in the number of purchases, under the appropriate centering and scaling, a Gaussian distribution is obtained across the entire phase. However, if the number of purchases is fixed, a degeneracy arises and normality holds only at the higher end of this phase. If the number of purchases have a fixed range, the small number of different coupons collected in the sublinear phase is upgraded to a number in need of centering and scaling to become normally distributed in the linear phase with a different normal distribution of the type that appears in the usual central limit theorems. The Gaussian results are obtained via martingale theory. We say a few words in passing about the high probability of collecting nearly all the coupons in the superlinear phase. It is our aim to present the results in a way that explores the critical transition at the ‘seam line’ between different Gaussian phases, and between these phases and other nonnormal phases.


2014 ◽  
Vol 1 ◽  
pp. 272-275 ◽  
Author(s):  
Vincent Canals ◽  
Antoni Morro ◽  
Josep L. Rosselló

2011 ◽  
Vol 36 (2) ◽  
pp. 251-266 ◽  
Author(s):  
Andrzej Rakowski ◽  
Piotr Rogowski

AbstractThis paper has two distinct parts. Section 1 includes general discussion of the phenomenon of "absolute pitch" (AP), and presentation of various concepts concerning definitions of "full", "partial" and "pseudo" AP. Sections 2-4 include presentation of the experiment concerning frequency range in which absolute pitch appears, and discussion of the experimental results. The experiment was performed with participation of 9 AP experts selected from the population of 250 music students as best scoring in the pitch-naming piano-tone screening tests. Each subject had to recognize chromas of 108 pure tones representing the chromatic musical scale of nine octaves from E0 to D#9. The series of 108 tones was presented to each subject 60 times in random order, diotically, with loudness level about 65 phon. Percentage of correct recognitions (PC) for each tone was computed. The frequency range for the existence of absolute pitch in pure tones, perceived by sensitive AP possessors stretches usually over 5 octaves from about 130.6 Hz (C3) to about 3.951 Hz (B7). However, it was noted that in a single case, the upper boundary of AP was 9.397 Hz (D9). The split-halves method was applied to estimate the reliability of the obtained results.


Author(s):  
Х.М. Хетагуров

Кленовники Северного Кавказа – особый тип лесных формаций. Особенность проявляется не только в структуре фитоценозов, но и в способах самовозобновления. Объект исследования – чистые и смешанные кленовники в нескольких урочищах РСО-Алания. Объекты расположены в различных условиях горного рельефа и на разной высоте над уровнем моря, от 1100 до 1900 м. Целью исследования является установление особенностей семенного возобновления кленовников, произрастающих в верхнем поясе распространения лесов на Северном Кавказе. Для достижения поставленной цели устанавливали время цветения и созревания семян по вертикальным поясам распространения кленовников. Обилие цветения и средний балл плодоношения устанавливали по В.Г. Капперу. Биометрические характеристики семян определяли по урочищам в трех повторностях. Учет подроста проводили на круговых учетных площадках по 10 м2, в соответствии с методикой А.В. Грязькина. Установлено, что цветение клена Траутфеттера начинается 10–14 мая и заканчивается в третьей декаде мая. Семена начинают созревать к концу августа – к началу сентября. Сроки опадения семян растянуты. Часть семян попадает на почву до начала листопада, поэтому они оказываются погребенными под опадом. Такие семена быстро теряют всхожесть, выпревают. Другая часть опадает вместе с листьями. Третья категория семян (самые легкие) держатся на деревьях до декабря и даже до весны. Естественное возобновление клена семенами под пологом материнского древостоя сильно затруднено из-за мощного травяного покрова. В древостое с сомкнутым пологом доля цветущих и плодоносящих деревьев составляет 2–3%, а средний балл плодоношения составляет 1,3. Освещенность на поверхности почвы не превышает 100 люкс, т. е. составляет 0,2–0,3% от освещенности на открытом месте; на высоте 1,3 м (над травостоем) освещенность несколько выше – от 1,2 до 1,6 тыс. люкс. Под пологом древостоя преобладает подрост вегетативного происхождения. У верхней границы кленового леса подрост семенного происхождения встречается в окнах, прогалинах и на открытых местах в количестве 430–630 экз./га. По высоте преобладает крупный подрост 50–60%, доля мелкого – 10–20%. The maple stands of the North Caucasus – a special type of forest formations. The peculiarity is manifested not only in the structure of phytocoenoses, but also in the ways of self-renewing. Object of research – pure and mixed maple stands in some areas of North Ossetia-Alania. The properties are located in various mountainous terrain and at different heights above sea level from 1100 to 1900 m. The aim of the study is to establish the characteristics of the seed the resumption of the maple stands growing in the upper belt of forests spread in the Northern Caucasus. To achieve this goal setting time of flowering and ripening of seeds on vertical zones distribution dominated. The abundance of flowering and the average score of fruiting was established by V.G. Capper. Biometric characteristics of the seeds was determined by the tracts in three replicates. Accounting for the undergrowth was carried out on a circular experimental plots at 10 m2, in accordance with the method of A.V. Grashkin. It is established that the flowering maple Trautvetter begins may 10–14 and ends in the third week of may. Seeds begin to ripen by late August – early September. The timing of subsidence of the stretched seed. Some of the seeds falls to the ground before the leaves, so they are trapped under the litter. These seeds quickly lose their germination, vypivaet. The other part falls along with the leaves. The third category of seeds (the light) stay on the trees until December and even till spring. Natural regeneration of maple seeds under the canopy of parent stand is very difficult due to the strong grass cover. In forest stands with dense canopy, proportion of flowering and fruit-bearing trees is 2–3%, and the average score of fruiting is 1.3. The illumination on the surface of the soil does not exceed 100 Lux, i. e. 0.2–0.3% of the light in the open, at a height of 1.3 m (on grass) the lighting is slightly higher, from 1.2 to 1.6 thousand Suite. Under the canopy of the forest the undergrowth is dominated by vegetative origin. At the upper boundary of the maple forest, saplings of seed origin occurs in Windows, clearings and open places in the number 430–630 ind./ha. In height dominated by large undergrowth of 50–60%, the share of small – 10–20%.


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