Segal's Burnside ring conjecture and the homotopy limit problem

AbstractPoisson processes in the space of $$(d-1)$$ ( d - 1 ) -dimensional totally geodesic subspaces (hyperplanes) in a d-dimensional hyperbolic space of constant curvature $$-1$$ - 1 are studied. The k-dimensional Hausdorff measure of their k-skeleton is considered. Explicit formulas for first- and second-order quantities restricted to bounded observation windows are obtained. The central limit problem for the k-dimensional Hausdorff measure of the k-skeleton is approached in two different set-ups: (i) for a fixed window and growing intensities, and (ii) for fixed intensity and growing spherical windows. While in case (i) the central limit theorem is valid for all $$d\ge 2$$ d ≥ 2 , it is shown that in case (ii) the central limit theorem holds for $$d\in \{2,3\}$$ d ∈ { 2 , 3 } and fails if $$d\ge 4$$ d ≥ 4 and $$k=d-1$$ k = d - 1 or if $$d\ge 7$$ d ≥ 7 and for general k. Also rates of convergence are studied and multivariate central limit theorems are obtained. Moreover, the situation in which the intensity and the spherical window are growing simultaneously is discussed. In the background are the Malliavin–Stein method for normal approximation and the combinatorial moment structure of Poisson U-statistics as well as tools from hyperbolic integral geometry.

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The central limit problem for trimmed sums

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100067359 ◽

1987 ◽

Vol 102 (2) ◽

pp. 329-349 ◽

Cited By ~ 23

Author(s):

Philip S. Griffin ◽

William E. Pruitt

Keyword(s):

Distribution Function ◽

Random Variables ◽

Random Variable ◽

Limit Problem ◽

Absolute Value ◽

Trimmed Sums ◽

Common Distribution ◽

Common Distribution Function ◽

Central Limit Problem ◽

Trimmed Sum

Let X, X1, X2,… be a sequence of non-degenerate i.i.d. random variables with common distribution function F. For 1 ≤ j ≤ n, let mn(j) be the number of Xi satisfying either |Xi| > |Xj|, 1 ≤ i ≤ n, or |Xi| = |Xj|, 1 ≤ i ≤ j, and let (r)Xn = Xj if mn(j) = r. Thus (r)Xn is the rth largest random variable in absolute value from amongst X1, …, Xn with ties being broken according to the order in which the random variables occur. Set (r)Sn = (r+1)Xn + … + (n)Xn and write Sn for (0)Sn. We will refer to (r)Sn as a trimmed sum.

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Stable cohomotopy and cobordism of abelian groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100058734 ◽

1981 ◽

Vol 90 (2) ◽

pp. 273-278 ◽

Cited By ~ 2

Author(s):

C. T. Stretch

Keyword(s):

Abelian Group ◽

Abelian Groups ◽

Finite Abelian Group ◽

Formal Group ◽

Characteristic Class ◽

Burnside Ring ◽

Classifying Space ◽

Formal Group Law ◽

Stable Cohomotopy ◽

Group Law

The object of this paper is to prove that for a finite abelian group G the natural map is injective, where Â(G) is the completion of the Burnside ring of G and σ0(BG) is the stable cohomotopy of the classifying space BG of G. The map â is detected by means of an M U* exponential characteristic class for permutation representations constructed in (11). The result is a generalization of a theorem of Laitinen (4) which treats elementary abelian groups using ordinary cohomology. One interesting feature of the present proof is that it makes explicit use of the universality of the formal group law of M U*. It also involves a computation of M U*(BG) in terms of the formal group law. This may be of independent interest. Since writing the paper the author has discovered that M U*(BG) has previously been calculated by Land-weber(5).

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Central Limit Problem

Probability Theory I - Graduate Texts in Mathematics ◽

10.1007/978-1-4757-6288-4_7 ◽

1977 ◽

pp. 280-352

Author(s):

M. Loève

Keyword(s):

Central Limit ◽

Limit Problem ◽

Central Limit Problem

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