Asymptotic Conditional Distribution of Exceedance Counts: Fragility Index with Different Margins

Let $\bm X=(X_1,...,X_d)$ be a random vector, whose components are not necessarily independent nor are they required to have identical distribution functions $F_1,...,F_d$. Denote by $N_s$ the number of exceedances among $X_1,...,X_d$ above a high threshold $s$. The fragility index, defined by $FI=\lim_{s\nearrow}E(N_s\mid N_s>0)$ if this limit exists, measures the asymptotic stability of the stochastic system $\bm X$ as the threshold increases. The system is called stable if $FI=1$ and fragile otherwise. In this paper we show that the asymptotic conditional distribution of exceedance counts (ACDEC) $p_k=\lim_{s\nearrow}P(N_s=k\mid N_s>0)$, $1\le k\le d$, exists, if the copula of $\bm X$ is in the domain of attraction of a multivariate extreme value distribution, and if $\lim_{s\nearrow}(1-F_i(s))/(1-F_\kappa(s))=\gamma_i\in[0,\infty)$ exists for $1\le i\le d$ and some $\kappa\in{1,...,d}$. This enables the computation of the FI corresponding to $\bm X$ and of the extended FI as well as of the asymptotic distribution of the exceedance cluster length also in that case, where the components of $\bm X$ are not identically distributed

An equivariant discrete model for complexified arrangement complements

We define a partial ordering on the set $\mathcal {Q}=\mathcal {Q}(\mathsf {M})$ of pairs of topes of an oriented matroid $\mathsf {M}$, and show the geometric realization $\vert\mathcal {Q}\vert$ of the order complex of $\mathcal {Q}$ has the same homotopy type as the Salvetti complex of $\mathsf {M}$. For any element $e$ of the ground set, the complex $\vert\mathcal {Q}_e\vert$ associated to the rank-one oriented matroid on $\{e\}$ has the homotopy type of the circle. There is a natural free simplicial action of $\mathbb{Z}_4$ on $\vert\mathcal {Q}\vert$, with orbit space isomorphic to the order complex of the poset $\mathcal {Q}(\mathsf {M},e)$ associated to the pointed (or affine) oriented matroid $(\mathsf {M},e)$. If $\mathsf {M}$ is the oriented matroid of an arrangement $\mathscr {A}$ of linear hyperplanes in $\mathbb{R}^n$, the $\mathbb{Z}_4$ action corresponds to the diagonal action of $\mathbb{C}^*$ on the complement $M$ of the complexification of $\mathscr {A}$: $\vert\mathcal {Q}\vert$ is equivariantly homotopy-equivalent to $M$ under the identification of $\mathbb{Z}_4$ with the multiplicative subgroup $\{\pm 1, \pm i\}\subset \mathbb{C}^*$, and $\vert\mathcal {Q}(\mathsf {M},e)\vert$ is homotopy- equivalent to the complement of the decone of $\mathscr {A}$ relative to the hyperplane corresponding to $e$. All constructions and arguments are carried out at the level of the underlying posets.We also show that the class of fundamental groups of such complexes is strictly larger than the class of fundamental groups of complements of complex hyperplane arrangements. Specifically, the group of the non- Pappus arrangement is not isomorphic to any realizable arrangement group. The argument uses new structural results concerning the degree-one resonance varieties of small matroids

Extracting ∣Vbc∣|V_{bc}|, mcm_c and mbm_b from Inclusive DD and BB Decays

Using recent results for nonperturbative contributions to the $B$ and $D$ meson inclusive semileptonic widths, a model independent extraction of \vbc, $m_c$ and $m_b$ is made from the experimentally measured $B$ and $D$ lifetimes and semileptonic branching ratios. Constraining the parameters of the HQET at \CO(1/m_Q^2) by the $D$ semileptonic width, \vbc is found to lie in the range .040<\vbc< 0.057. The $c$ and $b$ quark masses are not well constrained due to uncertainty in the relevant scale of $\alpha_s$. These results assume the validity of perturbative QCD at the low scales relevant to semileptonic charm decay. Without making this assumption, somewhat less stringent bounds on $V_{bc}$ from $B$ decay alone may be obtained.Comment: (revised version - contains a more detailed discussion of the uncertainty in our results from the uncertainty in the scale of \alpha_s) 12 pages, 5 figures included, uses harvmac.tex and epsf.tex, UCSD/PTH 93-25, UTPT 93-21, CMU-HEP 93-1