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Contemporary memory debates on representations of conflict and, in particular, crimes against humanity have undoubtedly been informed by the fairly recent emergence of perpetrator voices. As survivors of the Second World War reach the end of their lives, and as the event and related atrocities become increasingly distant in time, perpetrators are equally as aware as victims of a final opportunity to be heard directly. Paradoxically, too, war crimes trials since the 1990s, whilst intended to bring perpetrators to justice, have often provided them with a platform from which to 'talk back' to their victims, many of whom are no longer alive to defend themselves. Moreover, when groups of people associated with past oppression chose to remain silent (or were silenced), it is often the descendents of those implicated who publicly defend their cause today. Recent acts of genocide, notably in Rwanda and former Yugoslavia, the increasingly vocal demands for reparation from victims of genocide and massacres committed during the Second World War, and the 9/11 terrorist attacks on the USA have also played their part in defining the study of perpetrators and their misdeeds, and in shaping public and political debates.Academic scholars have lately sought to develop interpretative models for understanding and defining different kinds of 'extreme' violence, attempted to give meaning to specific acts of violence and questioned the ethical responsibilities involved in researching what is often a harrowing subject matter (see, for example, Sémelin et al., 2002). Whilst academic studies begin to stress the complexities surrounding perpetrator-hood and the ability of most human beings to become perpetrators under particular circumstances (see Bloxham and Kushner, 2005: 157-9), public reconstructions of the past according to victim-perpetrator/good-evil absolutes often fail to take account of the rather more blurred dynamics behind oppressive state rule and acts of atrocity.The construction and diffusion of memories of perpetrators and their crimes feed directly into, and are conditioned by, public debates relating to national/group identities, cultures and histories. Re-visitations of the past are inevitably conditioned by the imperative of national or group cohesion in the present. In the current international context of truth commissions, and restitution and compensation laws, in an age of commemorations and legislative deliberations relating to national histories, the apparent willingness of states to admit and make amends for past crimes against humanity has on occasion been contradicted by revisionist or 'normalizing' stances. In Germany,

1. The concentration function of a one-dimensional random variable X is defined to be ifj(x) = xjj{x: X) = supPr(a < X ^ a + x) where the supremum is taken over all real values of a. In & dimensions we can define the concentration function ifj(x) = tfj(x 1} x 2 , ...,x k ) in a similar way, taking the supremum over all ^-dimensional intervals whose sides are parallel to the coordinate axes. We can also define a 'spherical' concentration functionwhere this time the supremum is taken over all x 0 e R k and |. | is the usual Euclidean distance in R k .In this paper, we look for upper bounds for the concentration functions of sums of independent random variables, both in R x and in R k (k > 1). Many such inequalities have been given quite recently by Esseen in [6] and [7] but those which interest us here are in a way more primitive than his. We shall suppose that our random variables have moments up to the third order, and we shall look for upper bounds in terms of these moments and, in § 4, of simple properties of the densities when these exist.Inequalities involving the first three moments were among the first to be found for the concentration functions of sums of independent random variables. In [9] Theorem 1, Littlewood and Offord gave an upper bound for the concentration function of SjLi e r a r where the a r are given real or complex numbers and the e r are independent random variables taking the values + 1 and -1 each with probability \. Their inequality was sharpened by Erdos [4] and extended to sums of more general random variables by Offord [11]. Offord's theorem ([11] Theorem 1), with the notation altered, may be stated as follows. THEOREM A. Let X X ,X 2 , ...,X n be one-dimensional independent random variables with E(X r ) = 0, Math. Soc. (3) 23 (1971) 489-514 Proe. London

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1. Let ij, ri2 , ..., ij,, be independent identically distributed random variables in one dimension each having zero expectation and unit variance, let al , a2 , ..., a" be given real numbers and let S = alii1+a2i2+•••+annn•This note is concerned with an inequality for Pr (ISI > ACT) where Q2 = a1 2 +a2 2 +... +a. 2 and the emphasis is placed on large values of A. According to Tchebychev's inequality, this probability cannot exceed A -2 ; but naturally something so allembracing cannot be very sharp, and more can be said in particular cases. Thus if the rir have a normal (0, 1) distribution, S has a normal (0, a) distribution and it is easy to show from the corresponding distribution function that Pr (I SI > Au) < exp ( -+A2).An interesting inequality has been given by Khintchine [l; Hilfssatz 2]. If the rir take just the two values --1, +1 each with probability 4, he has shown that Pr (ISI > AQ) < C exp (-PAZ) where C is an absolute constant. Of course,t in this case S is bounded and so the probability is trivially zero if A is large enough, but there is a range of values of A where the inequality gives useful information not easily found in other ways. Thus if each ar is equal to 1, the largest value that ISI can take is n; since a = /n, the inequality shows that Pr (ISI > n) -(2M) , Ä2m ci) < .I (2e) e m•

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