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A Statistical Derivation of the Significant-Digit Law
The history, empirical evidence and classical explanations of the significant-digit (or Benford's) law are reviewed, followed by a sum- mary of recent invariant-measure characterizations. Then a newExpand
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Base-Invariance Implies Benford's Law
A derivation of Benford's Law or the First-Digit Phenomenon is given assuming only base-invariance of the underlying law. The only baseinvariant distributions are shown to be convex combinations ofExpand
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Comparisons of Stop Rule and Supremum Expectations of I.I.D. Random Variables
Implicitly defined (and easily approximated) universal constants 1.1 < an random variables and if Tn is the set of stop rules for Xl, "', Xn, then E(max{Xl , • • • ,Xn}) ~ an sup {EX, : tE Tn}, andExpand
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Strong laws for L- and U-statistics
Strong laws of large numbers are given for L-statistics (linear combinations of order statistics) and for U-statistics (averages of kernels of random samples) for ergodic stationary processes,Expand
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Prophet inequalities and order selection in optimal stopping problems
A complete determination is made of the possible values for E(sup X") and sup{ EX,: t a stop rule} for Xl, X2,... independent uniformly bounded random variables; this yields results of Krengel,Expand
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A Survey of Prophet Inequalities in Optimal Stopping Theory
This paper surveys the origin and development of what has come to be known as "prophet inequalities" in optimal stopping theory. Included is a review of all published work to date on these problems,Expand
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Partitioning General Probability Measures
On cherche a determiner des bornes de partition les meilleures possibles comme fonction de la taille maximum des atomes
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Stop rule inequalities for uniformly bounded sequences of random variables
If X0, XI..._ is an arbitrarily-dependent sequence of random variables taking values in [0, I] and if V( XA0, X,. . .) is the supremum, over stop rules t, of EX,, then the set of ordered pairs {(x,Expand
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Additive Comparisons of Stop Rule and Supremum Expectations of Uniformly Bounded Independent Random Variables
Let XI, X2, . . . be independent random variables taking values in [a, b], and let T denote the stop rules for X1, X2, Then E(sup,, X,,) sup{EX,: t E T) < (1/4)(b a), and this bound is best possible.Expand
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A basic theory of Benford's Law ∗
Drawing from a large, diverse body of work, this survey presents a comprehensive and unified introduction to the mathematics underlying the prevalent logarithmic distribution of significant digitsExpand
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