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\begin{document}
\bc{\large{\bf Explicit and asymptotic formulae for the expected values
of the order statistics of the Cantor distribution}}
Arnold Knopfmacher$^{\ft{\rm a}}$, Helmut Prodinger$^{\ft{\rm b}}$
{\em{\scriptsize${}^{{\rm a}}$Department of Computational \& Applied
Mathematics, Witwatersrand University, 2050, South Africa\\
${}^{\rm b}$Department of Algebra and Discrete Mathematics, Technical
University of Vienna, A-1040 Vienna, Austria}}\ec
\hrule
{\ft{\bf Abstract}
We derive the exact solution to a recurrence relation obtained by
Hosking for the expected value of the minimum order statistic of the
Cantor distribution. In addition, we indicate how an asymptotic
estimate can be derived for this and similar sums involving binomial
coefficients and Bernoulli numbers.
{\em Key words:} order statistics, alternating sums, Bernoulli numbers,
Rice's method}\\ [.3cm]
\hrule
\vspace{.5cm}
{\bf 1. Introduction}
\hspace{.6cm}In a recent article Hosking (1994) studied the order
statistics of the Cantor distribution. In particular he derived a
recurrence relation from which the average values
of the order statistics could be computed step by step.\\
\nh In this paper we derive an explicit solution to this recurence
relation as well as a simple asymptotic approximation to the exact
result. In doing this we indicate a general approach that could be used
to solve other recurrence relations of a similar nature.\\
\nh Rather than speaking of the Cantor distribution we will interpret
the recursion in terms of {\em random strings} $c_1\,c_2\,c_3\ldots$ where the
$c_i$'s $\in \{0,1\}$ are independently obtained by flips of a fair coin.
Given a parameter $\phi$ with $0<\phi\leq{1\over2}$ we then consider the
random variable $T$ that maps $c_1\,c_2\,c_3\ldots$ to the real number
$$
T(c_1\,c_2\,c_3\ldots)={1-\phi\over\phi}\;\sum_{i\geq1}c_i\phi^i\;
\in[0,1]\;.
$$
The strings now have a natural order from the usual ordering of the real
numbers. This is easily seen to be equivalent to the lexicographic
ordering of strings, i.e. $c_1\,c_2\,c_3\ldots