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Independent process approximations for random combinatorial structures
 ADVANCES IN MATHEMATICS
, 1994
"... Many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables, conditioned on the value of a weighted sum of the variables. It is interesting to compare the combinatorial structure directly to the independ ..."
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Cited by 38 (8 self)
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Many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables, conditioned on the value of a weighted sum of the variables. It is interesting to compare the combinatorial structure directly to the independent discrete process, without renormalizing. The quality of approximation can often be conveniently quantified in terms of total variation distance, for functionals which observe part, but not all, of the combinatorial and independent processes. Among the examples are combinatorial assemblies (e.g., permutations, random mapping functions, and partitions of a set), multisets (e.g, polynomials over a finite field, mapping patterns and partitions of an integer), and selections (e.g., partitions of an integer into distinct parts, and squarefree polynomials over finite fields). We consider issues common to all the above examples, including equalities and upper bounds for total variation distances, existence of limiting processes, heuristics for good approximations, the relation to standard generating functions, moment formulas and recursions for computing densities, refinement to the process which counts the number of parts of each possible type, the effect of further conditioning on events of moderate probability, large deviation theory and nonuniform measures on combinatorial objects, and the possibility of getting useful results by overpowering the conditioning.
Limit theorems for combinatorial structures via discrete process approximations
 RANDOM STRUCTURES AND ALGORITHMS
, 1992
"... Discrete functional limit theorems, which give independent process approximations for the joint distribution of the component structure of combinatorial objects such as permutations and mappings, have recently become available. In this article, we demonstrate the power of these theorems to provide e ..."
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Cited by 21 (2 self)
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Discrete functional limit theorems, which give independent process approximations for the joint distribution of the component structure of combinatorial objects such as permutations and mappings, have recently become available. In this article, we demonstrate the power of these theorems to provide elementary proofs of a variety of new and old limit theorems, including results previously proved by complicated analytical methods. Among the examples we treat are Brownian motion limit theorems for the cycle counts of a random permutation or the component counts of a random mapping, a Poisson limit law for the core of a random mapping, a generalization of the ErdosTurin Law for the logorder of a random permutation and the smallest component size of a random permutation, approximations to the joint laws of the smallest cycle sizes of a random mapping, and a limit distribution for the difference between the total number of cycles and the number of
On the amount of dependence in the prime factorization of a uniform random integer
, 2000
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Profiles of permutations
"... This paper develops an analogy between the cycle structure of, on the one hand, random permutations with cycle lengths restricted to lie in an infinite set S with asymptotic density σ and, on the other hand, permutations selected according to the Ewens distribution with parameter σ. In particular we ..."
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Cited by 4 (2 self)
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This paper develops an analogy between the cycle structure of, on the one hand, random permutations with cycle lengths restricted to lie in an infinite set S with asymptotic density σ and, on the other hand, permutations selected according to the Ewens distribution with parameter σ. In particular we show that the asymptotic expected number of cycles of random permutations of [n] with all cycles even, with all cycles odd, and chosen from the Ewens distribution with parameter 1/2 are all 1 2 log n + O(1), and the variance is of the same order. Furthermore, we show that in permutations of [n] chosen from the Ewens distribution with parameter σ, the probability of a random element being in a cycle longer than γn approaches (1−γ) σ for large n. The same limit law holds for permutations with cycles carrying multiplicative weights with average σ. We draw parallels between the Ewens distribution and the asymptoticdensity case and explain why these parallels should exist using permutations drawn from weighted Boltzmann distributions. 1
A study of counts of Bernoulli strings via conditional Poisson processes
, 2008
"... Abstract. A sequence of random variables, each taking values 0 or 1, is called a Bernoulli sequence. We say that a string of length d occurs, in a Bernoulli sequence, if a success is followed by exactly (d − 1) failures before the next success. The counts of such dstrings are of interest, and in sp ..."
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Cited by 3 (1 self)
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Abstract. A sequence of random variables, each taking values 0 or 1, is called a Bernoulli sequence. We say that a string of length d occurs, in a Bernoulli sequence, if a success is followed by exactly (d − 1) failures before the next success. The counts of such dstrings are of interest, and in specific independent Bernoulli sequences are known to correspond to asymptotic dcycle counts in random permutations. In this note, we give a new framework, in terms of conditional Poisson processes, which allows for a quick characterization of the joint distribution of the counts of all dstrings, in a general class of Bernoulli sequences, as certain mixtures of the product of Poisson measures. In particular, this general class includes all Bernoulli sequences considered in the literature, as well as a host of new sequences. 1.
The number of components in a logarithmic combinatorial structure
 Ann. Appl. Probab
, 2000
"... Under very mild conditions, we prove that the number of components in a decomposable logarithmic combinatorial structure has a distribution which is close to Poisson in total variation. The conditions are satisfied for all assemblies, multisets and selections in the logarithmic class. The error in t ..."
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Under very mild conditions, we prove that the number of components in a decomposable logarithmic combinatorial structure has a distribution which is close to Poisson in total variation. The conditions are satisfied for all assemblies, multisets and selections in the logarithmic class. The error in the Poisson approximation is shown under marginally more restrictive conditions to be of exact order O1 / log n, by exhibiting the penultimate asymptotic approximation; similar results have previously been obtained by Hwang [20], under stronger assumptions. Our method is entirely probabilistic, and the conditions can readily be verified in practice.
A Tale of Three Couplings: Poisson–Dirichlet and GEM Approximations for Random Permutations
, 2005
"... For a random permutation of n objects, as n → ∞, the process giving the proportion of elements in the longest cycle, the second longest cycle, and so on, converges in distribution to the PoissonDirichlet process with parameter 1. This was proved in 1977 by Kingman and by Vershik and Schmidt. For s ..."
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Cited by 2 (0 self)
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For a random permutation of n objects, as n → ∞, the process giving the proportion of elements in the longest cycle, the second longest cycle, and so on, converges in distribution to the PoissonDirichlet process with parameter 1. This was proved in 1977 by Kingman and by Vershik and Schmidt. For soft reasons, this is equivalent to the statement that the random permutations and the PoissonDirichlet process can be coupled so that zero is the limit of the expected ℓ1 distance between the process of cycle length proporortions and the PoissonDirichlet process. We investigate how rapid this metric convergence can be, and in doing so, give two new proofs of the distributional convergence. One of the couplings we consider has an analog for the prime factorizations of a uniformly distributed random integer, and these couplings rely on the “scale invariant spacing lemma” for the scale invariant Poisson processes, proved in this paper.
A probabilistic proof of the LindebergFeller central limit theorem Amer
 Math. Monthly
, 2009
"... The Central Limit Theorem, one of the most striking and useful results in probability and statistics, explains why the normal distribution appears in areas as diverse as gambling, measurement error, sampling, and statistical mechanics. In essence, the Central Limit Theorem states that the normal dis ..."
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The Central Limit Theorem, one of the most striking and useful results in probability and statistics, explains why the normal distribution appears in areas as diverse as gambling, measurement error, sampling, and statistical mechanics. In essence, the Central Limit Theorem states that the normal distribution
VolumeBased Morphometry of Brain MR Images in Adolescent Idiopathic Scoliosis and Healthy Control Subjects
"... BACKGROUND AND PURPOSE: Adolescent idiopathic scoliosis (AIS) is a spinal deformity with unknown cause. Previous studies have suggested that subclinical neurologic abnormalities are associated with AIS. The objective of this prospective study was to characterize systematically neuroanatomic changes ..."
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BACKGROUND AND PURPOSE: Adolescent idiopathic scoliosis (AIS) is a spinal deformity with unknown cause. Previous studies have suggested that subclinical neurologic abnormalities are associated with AIS. The objective of this prospective study was to characterize systematically neuroanatomic changes in patients with left thoracic AIS vs right thoracic AIS and healthy control subjects by using volumebased morphometry. MATERIALS AND METHODS: Our current study involved 9 girls with left thoracic AIS and 20 girls with right thoracic AIS vs 11 and 17 matched female control subjects, respectively. Voxelbased morphometry (VBM), deformationbased morphometry (DBM), and tensorbased morphometry (TBM) were used to analyze the MR images aligned with a specific brain template of local adolescent girls. The statistical t test was used in VBM and TBM, and the Hotelling T2 test was applied in DBM. RESULTS: Using VBM, we found statistically significant differences (P .05) in the white matter attenuation of the genu of the corpus callosum and left internal capsule (left thoracic AIS control subjects). In contrast, no significant differences were observed between patients with right thoracic AIS and control subjects. CONCLUSIONS: White matter attenuation in the corpus callosum and left internal capsule, responsible