Jet schemes and generating sequences of divisorial valuations in dimension two

AbstractCuller and Vogtmann defined a simplicial spaceO(g), calledouter space, to study the outer automorphism group of the free groupFg. Using representation theoretic methods, we give an embedding ofO(g) into the analytification of X(Fg,SL2(ℂ)), theSL2(ℂ) character variety ofFg, reproving a result of Morgan and Shalen. Then we show that every pointvcontained in a maximal cell ofO(g) defines a flat degeneration of X(Fg,SL2(ℂ)) to a toric varietyX(PΓ). We relate X(Fg,SL2(ℂ)) andX(v) topologically by showing that there is a surjective, continuous, proper map Ξv:X(Fg,SL2(ℂ)) →X(v). We then show that this map is a symplectomorphism on a dense open subset of X(Fg, SL2(ℂ)) with respect to natural symplectic structures on X(Fg, SL2(ℂ)) andX(v). In this way, we construct an integrable Hamiltonian system in X(Fg, SL2(ℂ)) for each point in a maximal cell ofO(g), and we show that eachvdefines a topological decomposition of X(Fg, SL2(ℂ)) derived from the decomposition ofX(PΓ) by its torus orbits. Finally, we show that the valuations coming from the closure of a maximal cell inO(g) all arise as divisorial valuations built from an associated projective compactification of X(Fg, SL2(ℂ)).

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First-order autoregressive gamma sequences and point processes

Advances in Applied Probability ◽

10.2307/1426429 ◽

1980 ◽

Vol 12 (3) ◽

pp. 727-745 ◽

Cited By ~ 196

Author(s):

D. P. Gaver ◽

P. A. W. Lewis

Keyword(s):

Parameter Estimation ◽

Point Processes ◽

Random Variables ◽

Innovation Process ◽

Sums Of Random Variables ◽

First Order ◽

Wide Range ◽

Serial Correlations ◽

Generating Sequences ◽

Stieltjes Transforms

It is shown that there is an innovation process {∊n} such that the sequence of random variables {Xn} generated by the linear, additive first-order autoregressive scheme Xn = pXn-1 + ∊n are marginally distributed as gamma (λ, k) variables if 0 ≦p ≦ 1. This first-order autoregressive gamma sequence is useful for modelling a wide range of observed phenomena. Properties of sums of random variables from this process are studied, as well as Laplace-Stieltjes transforms of adjacent variables and joint moments of variables with different separations. The process is not time-reversible and has a zero-defect which makes parameter estimation straightforward. Other positive-valued variables generated by the first-order autoregressive scheme are studied, as well as extensions of the scheme for generating sequences with given marginal distributions and negative serial correlations.

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First-order autoregressive gamma sequences and point processes

Advances in Applied Probability ◽

10.1017/s0001867800035473 ◽

1980 ◽

Vol 12 (03) ◽

pp. 727-745 ◽

Cited By ~ 35

Author(s):

D. P. Gaver ◽

P. A. W. Lewis

Keyword(s):

Parameter Estimation ◽

Point Processes ◽

Random Variables ◽

Innovation Process ◽

Sums Of Random Variables ◽

First Order ◽

Wide Range ◽

Serial Correlations ◽

Generating Sequences ◽

Stieltjes Transforms

It is shown that there is an innovation process {∊ n } such that the sequence of random variables {X n } generated by the linear, additive first-order autoregressive scheme X n = pXn-1 + ∊ n are marginally distributed as gamma (λ, k) variables if 0 ≦p ≦ 1. This first-order autoregressive gamma sequence is useful for modelling a wide range of observed phenomena. Properties of sums of random variables from this process are studied, as well as Laplace-Stieltjes transforms of adjacent variables and joint moments of variables with different separations. The process is not time-reversible and has a zero-defect which makes parameter estimation straightforward. Other positive-valued variables generated by the first-order autoregressive scheme are studied, as well as extensions of the scheme for generating sequences with given marginal distributions and negative serial correlations.

Download Full-text