Stirling Behavior is Asymptotically Normal

Harper, L. H.

doi:10.1214/aoms/1177698956

Cited by 271 publications

(182 citation statements)

References 2 publications

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“…The key idea is to write X P as a sum of independent random variables; the central limit theorem then follows, for example from the Berry-Esseen theorem. In the case in which all the η j are nonnegative the method goes back to Harper [14]. [4], To decompose X P as such a sum, we partition {1, .…”

Section: A Central Limit Theoremmentioning

confidence: 99%

“…Earlier Heilmann and Lieb [15] proved that all the zeros of the attendant matching polynomial P (z), whose coefficients p m enumerate incomplete matchings (monomer-dimer configurations) by the number m of edges (dimers), lie on the negative real axis. Harper [14] was the first to recognize-in a particular case of Stirling numbers-that such a property of a generating function P (z) meant that the distribution of the attendant random variable is one of a sum of independent, (0, 1)-valued, random variables; it instantly opened the door for his proof of asymptotic normality of those numbers. Godsil used Heilmann-Lieb's result and Harper's method to prove a CLT for {p m }, under a constraint on the ground graph guaranteeing that the variance tends to infinity.…”

Section: Introductionmentioning

confidence: 99%

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Central limit theorems, Lee–Yang zeros, and graph-counting polynomials

Lebowitz

Pittel

Ruelle

et al. 2016

Journal of Combinatorial Theory, Series A

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We consider the asymptotic normalcy of families of random variables X which count the number of occupied sites in some large set. We write Prob(X = m) = p m z m 0 /P (z 0 ), where P (z) is the generating function P (z) = N j=0 p j z j and z 0 > 0. We give sufficient criteria, involving the location of the zeros of P (z), for these families to satisfy a central limit theorem (CLT) and even a local CLT (LCLT); the theorems hold in the sense of estimates valid for large N (we assume that Var(X) is large when N is). For example, if all the zeros lie in the closed left half plane then X is asymptotically normal, and when the zeros satisfy some additional conditions then X satisfies an LCLT. We apply these results to cases in which X counts the number of edges in the (random) set of "occupied" edges in a graph, with constraints on the number of occupied edges attached to a given vertex. Our results also apply to systems of interacting particles, with X counting 1 the number of particles in a box Λ whose size |Λ| approaches infinity; P (z) is then the grand canonical partition function and its zeros are the Lee-Yang zeros.

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Section: A Central Limit Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Central limit theorems, Lee–Yang zeros, and graph-counting polynomials

Lebowitz

Pittel

Ruelle

et al. 2016

Journal of Combinatorial Theory, Series A

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show abstract

“…This is known to be true for partition lattices [9], til]. Another conjecture asserts that W. < W ■ whenever k .< r. 2 in a geometric lattice of rank r.…”

mentioning

confidence: 94%

Whitney Number Inequalities for Geometric Lattices

Dowling¹,

Wilson²

1975

Proceedings of the American Mathematical Society

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“…Jessen et al [4] showed that J c (k) ∼ k/log k as k → ∞. For c = 1, the asymptotics of J 1 (k) = J (k) were given earlier in [3], where a redundant e appeared in the denominator. Unfortunately, these asymptotics are extremely slow, as illustrated in Figure 1.…”

Section: Introductionmentioning

confidence: 94%

Asymptotics of conditional moments of the summand in Poisson compounds

Rolski¹,

Tomanek²

2011

Journal of Applied Probability

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Suppose that N is a Z + -valued random variable and that X, X 1 , X 2 , . . . is a sequence of independent and identically distributed Z + random variables independent of N . In this paper we are interested in properties of the conditional variable N kIn particular, we want to know the asymptotic behavior of the conditional mean E N k or the conditional variance var N k as k → ∞. We consider the cases when X is Poisson and when X is mixed Poisson. The problem is motivated by modeling loss reserves in nonlife insurance.

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Stirling Behavior is Asymptotically Normal

Cited by 271 publications

References 2 publications

Central limit theorems, Lee–Yang zeros, and graph-counting polynomials

Central limit theorems, Lee–Yang zeros, and graph-counting polynomials

Whitney Number Inequalities for Geometric Lattices

Asymptotics of conditional moments of the summand in Poisson compounds

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