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Mark Tushnet’s paper is an interesting extension of his recent work on how constitutional democracies can control the use of emergency powers. His analysis is useful in drawing attention to the often-overlooked impact of political dynamics in times of emergency. However, Tushnet is too ready to accept that political processes will be able to control the grant and use of emergency powers. He also is too quick to draw a sharp dichotomy between effective political and ineffective legal processes. Political processes can be active and useful in reining in the excesses of emergency powers, but usually only if they play out against the backdrop of a vibrant constitutional culture where the potential for abuse of emergency powers is recognised. The impact of legal and supranational controls plays an important role in shaping this constitutional culture, in particular by ‘dampening’ down the abuse of emergency powers and limiting the scope of action available to the executive. Focusing upon the potential of political means alone to control the application of state power in emergency states of exception is to underestimate the capacity of wider constitutional dynamics to play a key role, even if the political arena may be where the ultimate battles are fought out.
The issue of women's under-representation in politics continues to be controversial and pressing in countries all over the world. According to the Inter Parliamentary Union, only 14.7 per cent of the world's legislators are women.1 In no country do women achieve parity with men in terms of parliamentary representation. The record of many of the world's most developed countries is particularly poor. Looking at lower houses of the legislature, in the US women's representation is only 14 per cent, in the UK 18 per cent, France 12 per cent and Italy 10 per cent. These countries are considerably out-performed by others such as South Africa and Argentina, both at 30 per cent.2
We characterise the classes of continuous and discrete phase-type distributions in the following way. They are known to be closed under convolutions, mixtures, and the unary ‘geometric mixture' operation. We show that the continuous class is the smallest family of distributions that is closed under these operations and contains all exponential distributions and the point mass at zero. An analogous result holds for the discrete class.
We also show that discrete phase-type distributions can be regarded as ℝ+-rational sequences, in the sense of automata theory. This allows us to view our characterisation of them as a corollary of the Kleene–Schützenberger theorem on the behavior of finite automata. We prove moreover that any summable ℝ+-rational sequence is proportional to a discrete phase-type distribution.
The notion of an invariant polytope played a central role in the proof of the characterization of phase-type distributions. The purpose of this paper is to develop invariant polytope techniques further. We derive lower bounds on the number of states needed to represent a phase-type distribution based on poles of its Laplace–Stieltjes transform. We prove that every phase-type distribution whose transform has only real poles has a bidiagonal representation. We close with three short applications of the invariant polytope idea. Taken together, the results of this paper show that invariant polytopes provide a natural approach to many questions about phase-type distributions.
Given n independent, identically distributed random variables, let ρ n denote the probability that the maximum is unique. This probability is clearly unity if the distribution of the random variables is continuous. We explore the asymptotic behavior of the ρ n's in the case of geometric random variables. We find a function Φsuch that (ρ n – Φ(n)) → 0 as n →∞. In particular, we show that ρ n does not converge as n →∞. We derive a related asymptotic result for the expected value of the maximum of the sample. These results arose out of a random depletion model due to Bajaj, which was the original motivation for this paper and which is included.
The multivariate stochastic ordering induced by the convex nondecreasing functions compares a combination of size and variability of random vectors. Closely following methods developed by Strassen, we show that two probability measures are ordered in this way if and only if they are the marginals of some submartingale. The implications of this in majorisation theory are discussed.
A non-negative function f(t), t > 0, is said to be completely monotonic if its derivatives satisfy (-1)n fn (t) ≥ 0 for all t and n = 1, 2, …, For such a function, either f(t + δ) / f(t) is strictly increasing in t for each δ > 0, or f(t) = ce-dt for some constants c and d, and for all t. An application of this result is given.
The M/M/∞ queue can be considered as a compartmental system which may be modified by allowing its arrival and service rates to depend on an exogenous ‘environment process’. The states of the environment process are known as environments. We present some examples of this, chosen for their analytic or algorithmic tractability. We discuss (i) models with arrivals in only one environment, (ii) models with service in only one environment, (iii) reversible models and (iv) a model in which the environment process is also a queue.
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