189 research outputs found

    An improvement of the Berry--Esseen inequality with applications to Poisson and mixed Poisson random sums

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    By a modification of the method that was applied in (Korolev and Shevtsova, 2009), here the inequalities ρ(Fn,Φ)0.335789(β3+0.425)n\rho(F_n,\Phi)\le\frac{0.335789(\beta^3+0.425)}{\sqrt{n}} and ρ(Fn,Φ)0.3051(β3+1)n\rho(F_n,\Phi)\le \frac{0.3051(\beta^3+1)}{\sqrt{n}} are proved for the uniform distance ρ(Fn,Φ)\rho(F_n,\Phi) between the standard normal distribution function Φ\Phi and the distribution function FnF_n of the normalized sum of an arbitrary number n1n\ge1 of independent identically distributed random variables with zero mean, unit variance and finite third absolute moment β3\beta^3. The first of these inequalities sharpens the best known version of the classical Berry--Esseen inequality since 0.335789(β3+0.425)0.335789(1+0.425)β3<0.4785β30.335789(\beta^3+0.425)\le0.335789(1+0.425)\beta^3<0.4785\beta^3 by virtue of the condition β31\beta^3\ge1, and 0.4785 is the best known upper estimate of the absolute constant in the classical Berry--Esseen inequality. The second inequality is applied to lowering the upper estimate of the absolute constant in the analog of the Berry--Esseen inequality for Poisson random sums to 0.3051 which is strictly less than the least possible value of the absolute constant in the classical Berry--Esseen inequality. As a corollary, the estimates of the rate of convergence in limit theorems for compound mixed Poisson distributions are refined.Comment: 33 page

    'Drowning in here in his bloody sea' : exploring TV cop drama's representations of the impact of stress in modern policing

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    The Criminal Justice System is a part of society that is both familiar and hidden. It is familiar in that a large part of daily news and television drama is devoted to it (Carrabine, 2008; Jewkes, 2011). It is hidden in the sense that the majority of the population have little, if any, direct contact with the Criminal Justice System, meaning that the media may be a major force in shaping their views on crime and policing (Carrabine, 2008). As Reiner (2000) notes, the debate about the relationship between the media, policing, and crime has been a key feature of wider societal concerns about crime since the establishment of the modern police force. He outlines the recurring themes in post-war debates in this field. For Conservatives there has been an ongoing concern that the media is criminongenic, as it serves to undermine traditional institutions, including the police. From the viewpoint of radical criminology, the impact of the media is two-fold: it exaggerates legitimate concerns about crime and emphasises the bureaucratic and other restrictions under which the police operate (Reiner, 2000). This is seen as undermining due process and legitimatising what can be termed a ‘maverick’ approach to policing. An early example of this can be seen in Clint Eastwood’s Dirty Harry movies (Siegel, 1971) where Harry Callaghan acts as a one-man law enforcement system outside of the formal legal process, a process portrayed as corrupt, inefficient, and concerned with offenders’ rights rather than protecting victims. From a policing perspective, Reiner (2000) argues that film and TV drama creates a simplistic narrative of crime solving that is almost completely divorced from the reality of modern police work, a finding consistent with more recent work by Cummins et al., (2014)
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