Pub Date : 1998-10-01DOI: 10.1002/(SICI)1098-2418(199810/12)13:3/4%3C467::AID-RSA15%3E3.0.CO;2-W
S. Janson
1. Introduction Suen 8] found a remarkable correlation inequality, giving estimates for the probability that a collection of dependent random indicator variables vanish simultaneously, or in other words, for the probability that none of a collection of dependent events occurs. The present author 4, 3] has found similar inequalities for a much more restricted situation; when applicable, these inequalities are somewhat better than Suen's, although the diierence is negligible in many cases. (See Section 8 below.) Those inequalities have been used by several diierent authors for a variety of problems; there are, however, many situations where they are not applicable (see 8, 5] for two examples) and then Suen's inequality is a very attractive choice. The purpose of the present note is to present some improvements and mod-iications of Suen's original inequality which (we hope) will be easy to apply in diierent situations. The estimates considered here are exponential (unlike for example Cheby-shev's inequality), in the sense that they typically are similar to the estimate exp() for the independent case, where is the expected number of events. They are thus aimed at the case when the studied probability is very small, and has to be shown to be very small. In many applications, constants oc-curing in the estimates, even in the exponents, are immaterial; on the other hand, there are applications where very precise estimates are desired. For this reason, and because diierent versions of the inequality turn out to be useful in diierent situations, we will give several diierent versions of our estimates. We give several upper bounds to the probability of simultaneous vanishing of a collection of indicator variables in Section 3; these are perhaps the main results of the paper. We give some corresponding lower bounds in Section 4, and in Section 5 an upper bound for the probability that only a few of the variables are non-zero. Section 6 contains the proofs of the results, while Section 7 contains three examples related to the sharpness of the results. Finally, Section 8 contains a short discussion of the results and some open problems. Acknowledgement. This paper has beneetted from discussions with par
{"title":"New versions of Suen's correlation inequality","authors":"S. Janson","doi":"10.1002/(SICI)1098-2418(199810/12)13:3/4%3C467::AID-RSA15%3E3.0.CO;2-W","DOIUrl":"https://doi.org/10.1002/(SICI)1098-2418(199810/12)13:3/4%3C467::AID-RSA15%3E3.0.CO;2-W","url":null,"abstract":"1. Introduction Suen 8] found a remarkable correlation inequality, giving estimates for the probability that a collection of dependent random indicator variables vanish simultaneously, or in other words, for the probability that none of a collection of dependent events occurs. The present author 4, 3] has found similar inequalities for a much more restricted situation; when applicable, these inequalities are somewhat better than Suen's, although the diierence is negligible in many cases. (See Section 8 below.) Those inequalities have been used by several diierent authors for a variety of problems; there are, however, many situations where they are not applicable (see 8, 5] for two examples) and then Suen's inequality is a very attractive choice. The purpose of the present note is to present some improvements and mod-iications of Suen's original inequality which (we hope) will be easy to apply in diierent situations. The estimates considered here are exponential (unlike for example Cheby-shev's inequality), in the sense that they typically are similar to the estimate exp() for the independent case, where is the expected number of events. They are thus aimed at the case when the studied probability is very small, and has to be shown to be very small. In many applications, constants oc-curing in the estimates, even in the exponents, are immaterial; on the other hand, there are applications where very precise estimates are desired. For this reason, and because diierent versions of the inequality turn out to be useful in diierent situations, we will give several diierent versions of our estimates. We give several upper bounds to the probability of simultaneous vanishing of a collection of indicator variables in Section 3; these are perhaps the main results of the paper. We give some corresponding lower bounds in Section 4, and in Section 5 an upper bound for the probability that only a few of the variables are non-zero. Section 6 contains the proofs of the results, while Section 7 contains three examples related to the sharpness of the results. Finally, Section 8 contains a short discussion of the results and some open problems. Acknowledgement. This paper has beneetted from discussions with par","PeriodicalId":303496,"journal":{"name":"Random Struct. Algorithms","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131131921","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1998-10-01DOI: 10.1002/(SICI)1098-2418(199810/12)13:3/4%3C383::AID-RSA10%3E3.0.CO;2-0
Bernd Kreuter
{"title":"Small sublattices in random subsets of Boolean lattices","authors":"Bernd Kreuter","doi":"10.1002/(SICI)1098-2418(199810/12)13:3/4%3C383::AID-RSA10%3E3.0.CO;2-0","DOIUrl":"https://doi.org/10.1002/(SICI)1098-2418(199810/12)13:3/4%3C383::AID-RSA10%3E3.0.CO;2-0","url":null,"abstract":"","PeriodicalId":303496,"journal":{"name":"Random Struct. Algorithms","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123499652","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1998-10-01DOI: 10.1002/(SICI)1098-2418(199810/12)13:3/4%3C189::AID-RSA1%3E3.0.CO;2-R
A. Panholzer, H. Prodinger
Hoare’s FIND algorithm can be used to select p specified order statistics n Ž . 4 j , . . . , j from a file of n elements simultaneously. Averaging over all subsets of 1, . . . , n p 1 p defines the grand a ̈erages of the number of passes and comparisons. We use a generating functions approach to compute averages and variances of these parameters. Additionally, we sketch analogous developments for the instance of median-of-three partition and binary Ž . Catalan trees. Q 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13, 189]209, 1998
Hoare的FIND算法可用于选择p指定阶统计量n Ž。4、……同时从n个元素的文件中取出j。对1，…的所有子集求平均值。， n p 1 p定义了传递和比较次数的大平均值。我们使用生成函数方法来计算这些参数的平均值和方差。此外，我们概述了类似的开发实例的中位数的三个分区和二进制Ž。加泰罗尼亚的树木。John Wiley & Sons, Inc.;随机结构。Alg。[j]中国农业科学，1999,19 (2):1 - 4
{"title":"A generating functions approach for the analysis of grand averages for multiple QUICKSELECT","authors":"A. Panholzer, H. Prodinger","doi":"10.1002/(SICI)1098-2418(199810/12)13:3/4%3C189::AID-RSA1%3E3.0.CO;2-R","DOIUrl":"https://doi.org/10.1002/(SICI)1098-2418(199810/12)13:3/4%3C189::AID-RSA1%3E3.0.CO;2-R","url":null,"abstract":"Hoare’s FIND algorithm can be used to select p specified order statistics n Ž . 4 j , . . . , j from a file of n elements simultaneously. Averaging over all subsets of 1, . . . , n p 1 p defines the grand a ̈erages of the number of passes and comparisons. We use a generating functions approach to compute averages and variances of these parameters. Additionally, we sketch analogous developments for the instance of median-of-three partition and binary Ž . Catalan trees. Q 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13, 189]209, 1998","PeriodicalId":303496,"journal":{"name":"Random Struct. Algorithms","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130623207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1998-10-01DOI: 10.1002/(SICI)1098-2418(199810/12)13:3/4%3C211::AID-RSA2%3E3.0.CO;2-Y
L. Mutafchiev
We consider four families of forests on n vertices: labeled and unlabeled forests containing rooted and unrooted trees, respectively. A forest is chosen uniformly from one of the given four families. The limiting distribution of the size of its largest tree is then studied as na`. Convergences to discrete distributions depending on 1r2and 3r2-stable probability densities are established. It turns out that 1r2-stability parameter appears when Ž the random forest consists of rooted trees only. Otherwise i.e., when it contains only . unrooted trees , this parameter is 3r2. Furthermore, we show that the labels’ deletion of forest’s vertices does not change the corresponding limiting law in general; it changes the values of some scaling and additive parameters of the limiting distributions that we obtain. Q 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13, 211]228, 1998
{"title":"The largest tree in certain models of random forests","authors":"L. Mutafchiev","doi":"10.1002/(SICI)1098-2418(199810/12)13:3/4%3C211::AID-RSA2%3E3.0.CO;2-Y","DOIUrl":"https://doi.org/10.1002/(SICI)1098-2418(199810/12)13:3/4%3C211::AID-RSA2%3E3.0.CO;2-Y","url":null,"abstract":"We consider four families of forests on n vertices: labeled and unlabeled forests containing rooted and unrooted trees, respectively. A forest is chosen uniformly from one of the given four families. The limiting distribution of the size of its largest tree is then studied as na`. Convergences to discrete distributions depending on 1r2and 3r2-stable probability densities are established. It turns out that 1r2-stability parameter appears when Ž the random forest consists of rooted trees only. Otherwise i.e., when it contains only . unrooted trees , this parameter is 3r2. Furthermore, we show that the labels’ deletion of forest’s vertices does not change the corresponding limiting law in general; it changes the values of some scaling and additive parameters of the limiting distributions that we obtain. Q 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13, 211]228, 1998","PeriodicalId":303496,"journal":{"name":"Random Struct. Algorithms","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117295217","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1998-10-01DOI: 10.1002/(SICI)1098-2418(199810/12)13:3/4%3C319::AID-RSA7%3E3.0.CO;2-Y
Z. Bai, Hsien-Kuei Hwang, Wen-Qi Liang
We establish the asymptotic normality of the number of upper records in a sequence of iid geometric random variables. Large deviations and local limit theorems as well as approximation theorems for the number of lower records are also derived.
{"title":"Normal approximations of the number of records in geometrically distributed random variables","authors":"Z. Bai, Hsien-Kuei Hwang, Wen-Qi Liang","doi":"10.1002/(SICI)1098-2418(199810/12)13:3/4%3C319::AID-RSA7%3E3.0.CO;2-Y","DOIUrl":"https://doi.org/10.1002/(SICI)1098-2418(199810/12)13:3/4%3C319::AID-RSA7%3E3.0.CO;2-Y","url":null,"abstract":"We establish the asymptotic normality of the number of upper records in a sequence of iid geometric random variables. Large deviations and local limit theorems as well as approximation theorems for the number of lower records are also derived.","PeriodicalId":303496,"journal":{"name":"Random Struct. Algorithms","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128920778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1998-10-01DOI: 10.1002/(SICI)1098-2418(199810/12)13:3/4%3C409::AID-RSA11%3E3.0.CO;2-U
J. Balogh, G. Pete
We introduce some generalizations of a nice combinatorial problem, the central notion of which is the so-called Disease Process. Let us color independently each square of an n×n chessboard black with a probability p(n), this is a random initial configuration of our process. Then we have a deterministic painting or expansion rule, and the question is the behaviour of the disease process determined by this rule of spreading. In particular, how large p(n) must be for painting the whole chessboard black? The main result of this paper is the almost exact determination of the threshold function in the fundamental case of this Random Disease Problem. Further investigations are involved about the general randomized and deterministic cases.
{"title":"Random disease on the square grid","authors":"J. Balogh, G. Pete","doi":"10.1002/(SICI)1098-2418(199810/12)13:3/4%3C409::AID-RSA11%3E3.0.CO;2-U","DOIUrl":"https://doi.org/10.1002/(SICI)1098-2418(199810/12)13:3/4%3C409::AID-RSA11%3E3.0.CO;2-U","url":null,"abstract":"We introduce some generalizations of a nice combinatorial problem, the central notion of which is the so-called Disease Process. Let us color independently each square of an n×n chessboard black with a probability p(n), this is a random initial configuration of our process. Then we have a deterministic painting or expansion rule, and the question is the behaviour of the disease process determined by this rule of spreading. In particular, how large p(n) must be for painting the whole chessboard black? The main result of this paper is the almost exact determination of the threshold function in the fundamental case of this Random Disease Problem. Further investigations are involved about the general randomized and deterministic cases.","PeriodicalId":303496,"journal":{"name":"Random Struct. Algorithms","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116797473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：