SIAM Journal on Matrix Analysis and Applications, 2011
A sequence of real numbers (x n ) is Benford if the significands, i.e. the fraction parts in the ... more A sequence of real numbers (x n ) is Benford if the significands, i.e. the fraction parts in the floating-point representation of (x n ), are distributed logarithmically. Similarly, a discrete-time irreducible and aperiodic finite-state Markov chain with transition probability matrix P and limiting matrix P * is Benford if every component of both sequences of matrices (P n − P * ) and (P n+1 − P n ) is Benford or eventually zero. Using recent tools that established Benford behavior for finite-dimensional linear maps, via the classical theories of uniform distribution modulo 1 and Perron-Frobenius, this paper derives a simple sufficient condition ("nonresonance") guaranteeing that P , or the Markov chain associated with it, is Benford. This result in turn is used to show that almost all Markov chains are Benford, in the sense that if the transition probability matrix is chosen in an absolutely continuous manner, then the resulting Markov chain is Benford with probability one. Concrete examples illustrate the various cases that arise, and the theory is complemented with simulations and potential applications.
For the best-choice (or secretary) problem with an unknown number N of objects, minimax-optimal s... more For the best-choice (or secretary) problem with an unknown number N of objects, minimax-optimal strategies for the observer and minimax distributions for N are derived under the assumption that N is a random variable with expected value at most M, where M is known. The solution is derived as a special case of the situation where N is constrained by Ef(N) <_ M, where f is increasing with f(i) f(i 1) convex.
Feller's classic text 'An Introduction to Probability Theory and its Applications' contains a der... more Feller's classic text 'An Introduction to Probability Theory and its Applications' contains a derivation of the well known significant-digit law called Benford's law. More specifically, Feller gives a sufficient condition ("large spread") for a random variable $X$ to be approximately Benford distributed, that is, for $\log_{10}X$ to be approximately uniformly distributed modulo one. This note shows that the large-spread derivation, which continues to be widely cited and used, contains serious basic errors. Concrete examples and a new inequality clearly demonstrate that large spread (or large spread on a logarithmic scale) does not imply that a random variable is approximately Benford distributed, for any reasonable definition of "spread" or measure of dispersion
For the secretary (or best-choice) problem with an unknown number N of objects, minimax-optimal s... more For the secretary (or best-choice) problem with an unknown number N of objects, minimax-optimal stop rules and (worst-case) distributions are derived, under the assumption that N is a random variable with unknown distribution, but known upper bound n. Asymptotically, the probability of selecting the best object in this situation is of order of (log n)-Y. For example, even if the only information available is that there are somewhere between 1 and 100 objects, there is still a strategy which will select the best item about one time in five.
Nachrichten Der Akademie Der Wissenschaften in Gottingen 2 Mathematisch Physikalische Klasse, 2002
L evy's classical continuity theorem states that if the pointwise limit of a sequence of cha... more L evy's classical continuity theorem states that if the pointwise limit of a sequence of characteristic functions exists, then the limit function itself is a characteristic function if and only if the limit function satises a single universal limit condition (in his case, the limit at zero is one), in which case the underlying measures converge weakly to the probability measure represented by the limit function. It is the purpose of this article to give a number of direct analogs of L evy's theorem for other probability-representing functions including moment sequences, maximal moment sequences, mean-residual-life functions, Hardy-Littlewood maximal functions, and failure-rate functions. In each of these cases the single crucial condition on the limit function often relates to conservation of mass or moment, but a general theory encompassing all of these examples is still missing.
The formal mathematical theory of fair division has a rich history dating back at least to Steinh... more The formal mathematical theory of fair division has a rich history dating back at least to Steinhaus in the 1940's. In recent work in this area, several general classes of errors have appeared along with confusion about the necessity and sufficiency of certain hypotheses. It is the purpose of this article to correct the scientific record and to point out with concrete examples some of the pitfalls that have led to these mistakes. These examples may serve as guideposts for future work.
Strong laws of large numbers are given for L-statistics (linear combinations of order statistics)... more 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, extending classical theorems of Hoeffding and of Helmers for iid sequences. Examples are given to show that strong and even weak convergence may fail if the given sufficient conditions are not satisfied, and an application is given to estimation of correlation dimension of invariant measures.
Sharp lower bounds are found for the concentration of a probability distribution as a function of... more Sharp lower bounds are found for the concentration of a probability distribution as a function of the expectation of any given convex symmetric function¢. In the case ¢(x)=(x-c)2, where c is the expected value of the distribution, these bounds yield the classical concentration-variance inequality of Levy. An analogous sharp inequality is obtained in a similar linear search setting, where a sharp lower bound for the concentration is found as a function of the maximum probability swept out from a fixed starting point by a path of given length.
Near a stable xed point at 0 or1, many real-valued dynamical systems follow Benford's law: un... more Near a stable xed point at 0 or1, many real-valued dynamical systems follow Benford's law: under iteration of a map T the proportion of values infx;T (x);T2(x);:::;Tn(x)g with mantissa (base b )l ess thant tends to logbt for all t in (1;b )a sn!1 ,f or all integer bases b> 1. In particu- lar, the orbits under most power, exponential,
Strong laws of large numbers are given forL-statistics (linear com- binations of order statistics... more Strong laws of large numbers are given forL-statistics (linear com- binations of order statistics) and for U-statistics (averages of kernels of ran- dom samples) for ergodic stationary processes, extending classical theorems of Hoeding and of Helmers for iid sequences. Examples are given to show that strong and even weak convergence may fail if the given sucient conditions are not satised, and an application is given to estimation of correlation dimension of invariant measures.
Transactions of the American Mathematical Society, 1983
If X"0, Xx_ is an arbitrarily-dependent sequence of random variables taking values in [0,1] and i... more If X"0, Xx_ is an arbitrarily-dependent sequence of random variables taking values in [0,1] and if V( X0,X¡,... ) is the supremum, over stop rules /, of EX,, then the set of ordered pairs {(.*, v)
Transactions of the American Mathematical Society, 1986
Suppose fix,... ,fin are nonatomic probability measures on the same measurable space (S, S). Then... more Suppose fix,... ,fin are nonatomic probability measures on the same measurable space (S, S). Then there exists a measurable partition isi}"=i of 5 such that Pi(Si) > (n + 1 M)'1 for a11 i l,...,n, where M is the total mass of V?=i ĂŸi (tne smallest measure majorizing each m). This inequality is the best possible for the functional M, and sharpens and quantifies a well-known cake-cutting theorem of Urbanik and of Dubins and Spanier. Applications are made to L\ -functions, discrete allocation problems, statistical decision theory, and a dual problem.
;dly all the classical dynamic programming expected pnyolf function.\. reward function as well, i... more ;dly all the classical dynamic programming expected pnyolf function.\. reward function as well, in contrast to the usual randomized Markov average which depends only on the original strategy [7, 8, 11, 14, 191.
A class of empirical distributions is introduced which are based on various weighted linear combi... more A class of empirical distributions is introduced which are based on various weighted linear combinations of order statistics, and which have convergence properties the classical empirical distribution does not, or which stochastically or convexly dominate the classical empirical distribution
L evy's classical continuity theorem states that if the pointwise limit of a sequence of cha... more L evy's classical continuity theorem states that if the pointwise limit of a sequence of characteristic functions exists, then the limit function itself is a characteristic function if and only if the limit function satises a single universal limit condition (in his case, the limit at zero is one), in which case the underlying measures converge weakly to the probability measure represented by the limit function. It is the purpose of this article to give a number of direct analogs of L evy's theorem for other probability-representing functions including moment sequences, maximal moment sequences, mean-residual-life functions, Hardy-Littlewood maximal functions, and failure-rate functions. In each of these cases the single crucial condition on the limit function often relates to conservation of mass or moment, but a general theory encompassing all of these examples is still missing.
Proceedings of the American Mathematical Society, 1994
It is well known that the expected values {Mk(X)}, k > 1, of the k-maximal order statistics of an... more It is well known that the expected values {Mk(X)}, k > 1, of the k-maximal order statistics of an integrable random variable X uniquely determine the distribution of X. The main result in this paper is that if {X, }, n > 1 , is a sequence of integrable random variables with lime -00 Mk (Xn) = ak for all k > 1 , then there exists a random variable X with Mk(X) = ak for all k > 1 and Xn
Proceedings of the American Mathematical Society, 1981
Let XI, X2, . . . be independent random variables taking values in [a, b], and let T denote the s... more 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)(ba), and this bound is best possible. Probabilistically, this says that if a prophet (player with complete foresight) makes a side payment of (ba)/8 to a gambler (player using nonanticipating stop rules), the game becomes at least fair for the gambler.
SIAM Journal on Matrix Analysis and Applications, 2011
A sequence of real numbers (x n ) is Benford if the significands, i.e. the fraction parts in the ... more A sequence of real numbers (x n ) is Benford if the significands, i.e. the fraction parts in the floating-point representation of (x n ), are distributed logarithmically. Similarly, a discrete-time irreducible and aperiodic finite-state Markov chain with transition probability matrix P and limiting matrix P * is Benford if every component of both sequences of matrices (P n − P * ) and (P n+1 − P n ) is Benford or eventually zero. Using recent tools that established Benford behavior for finite-dimensional linear maps, via the classical theories of uniform distribution modulo 1 and Perron-Frobenius, this paper derives a simple sufficient condition ("nonresonance") guaranteeing that P , or the Markov chain associated with it, is Benford. This result in turn is used to show that almost all Markov chains are Benford, in the sense that if the transition probability matrix is chosen in an absolutely continuous manner, then the resulting Markov chain is Benford with probability one. Concrete examples illustrate the various cases that arise, and the theory is complemented with simulations and potential applications.
For the best-choice (or secretary) problem with an unknown number N of objects, minimax-optimal s... more For the best-choice (or secretary) problem with an unknown number N of objects, minimax-optimal strategies for the observer and minimax distributions for N are derived under the assumption that N is a random variable with expected value at most M, where M is known. The solution is derived as a special case of the situation where N is constrained by Ef(N) <_ M, where f is increasing with f(i) f(i 1) convex.
Feller's classic text 'An Introduction to Probability Theory and its Applications' contains a der... more Feller's classic text 'An Introduction to Probability Theory and its Applications' contains a derivation of the well known significant-digit law called Benford's law. More specifically, Feller gives a sufficient condition ("large spread") for a random variable $X$ to be approximately Benford distributed, that is, for $\log_{10}X$ to be approximately uniformly distributed modulo one. This note shows that the large-spread derivation, which continues to be widely cited and used, contains serious basic errors. Concrete examples and a new inequality clearly demonstrate that large spread (or large spread on a logarithmic scale) does not imply that a random variable is approximately Benford distributed, for any reasonable definition of "spread" or measure of dispersion
For the secretary (or best-choice) problem with an unknown number N of objects, minimax-optimal s... more For the secretary (or best-choice) problem with an unknown number N of objects, minimax-optimal stop rules and (worst-case) distributions are derived, under the assumption that N is a random variable with unknown distribution, but known upper bound n. Asymptotically, the probability of selecting the best object in this situation is of order of (log n)-Y. For example, even if the only information available is that there are somewhere between 1 and 100 objects, there is still a strategy which will select the best item about one time in five.
Nachrichten Der Akademie Der Wissenschaften in Gottingen 2 Mathematisch Physikalische Klasse, 2002
L evy's classical continuity theorem states that if the pointwise limit of a sequence of cha... more L evy's classical continuity theorem states that if the pointwise limit of a sequence of characteristic functions exists, then the limit function itself is a characteristic function if and only if the limit function satises a single universal limit condition (in his case, the limit at zero is one), in which case the underlying measures converge weakly to the probability measure represented by the limit function. It is the purpose of this article to give a number of direct analogs of L evy's theorem for other probability-representing functions including moment sequences, maximal moment sequences, mean-residual-life functions, Hardy-Littlewood maximal functions, and failure-rate functions. In each of these cases the single crucial condition on the limit function often relates to conservation of mass or moment, but a general theory encompassing all of these examples is still missing.
The formal mathematical theory of fair division has a rich history dating back at least to Steinh... more The formal mathematical theory of fair division has a rich history dating back at least to Steinhaus in the 1940's. In recent work in this area, several general classes of errors have appeared along with confusion about the necessity and sufficiency of certain hypotheses. It is the purpose of this article to correct the scientific record and to point out with concrete examples some of the pitfalls that have led to these mistakes. These examples may serve as guideposts for future work.
Strong laws of large numbers are given for L-statistics (linear combinations of order statistics)... more 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, extending classical theorems of Hoeffding and of Helmers for iid sequences. Examples are given to show that strong and even weak convergence may fail if the given sufficient conditions are not satisfied, and an application is given to estimation of correlation dimension of invariant measures.
Sharp lower bounds are found for the concentration of a probability distribution as a function of... more Sharp lower bounds are found for the concentration of a probability distribution as a function of the expectation of any given convex symmetric function¢. In the case ¢(x)=(x-c)2, where c is the expected value of the distribution, these bounds yield the classical concentration-variance inequality of Levy. An analogous sharp inequality is obtained in a similar linear search setting, where a sharp lower bound for the concentration is found as a function of the maximum probability swept out from a fixed starting point by a path of given length.
Near a stable xed point at 0 or1, many real-valued dynamical systems follow Benford's law: un... more Near a stable xed point at 0 or1, many real-valued dynamical systems follow Benford's law: under iteration of a map T the proportion of values infx;T (x);T2(x);:::;Tn(x)g with mantissa (base b )l ess thant tends to logbt for all t in (1;b )a sn!1 ,f or all integer bases b> 1. In particu- lar, the orbits under most power, exponential,
Strong laws of large numbers are given forL-statistics (linear com- binations of order statistics... more Strong laws of large numbers are given forL-statistics (linear com- binations of order statistics) and for U-statistics (averages of kernels of ran- dom samples) for ergodic stationary processes, extending classical theorems of Hoeding and of Helmers for iid sequences. Examples are given to show that strong and even weak convergence may fail if the given sucient conditions are not satised, and an application is given to estimation of correlation dimension of invariant measures.
Transactions of the American Mathematical Society, 1983
If X"0, Xx_ is an arbitrarily-dependent sequence of random variables taking values in [0,1] and i... more If X"0, Xx_ is an arbitrarily-dependent sequence of random variables taking values in [0,1] and if V( X0,X¡,... ) is the supremum, over stop rules /, of EX,, then the set of ordered pairs {(.*, v)
Transactions of the American Mathematical Society, 1986
Suppose fix,... ,fin are nonatomic probability measures on the same measurable space (S, S). Then... more Suppose fix,... ,fin are nonatomic probability measures on the same measurable space (S, S). Then there exists a measurable partition isi}"=i of 5 such that Pi(Si) > (n + 1 M)'1 for a11 i l,...,n, where M is the total mass of V?=i ĂŸi (tne smallest measure majorizing each m). This inequality is the best possible for the functional M, and sharpens and quantifies a well-known cake-cutting theorem of Urbanik and of Dubins and Spanier. Applications are made to L\ -functions, discrete allocation problems, statistical decision theory, and a dual problem.
;dly all the classical dynamic programming expected pnyolf function.\. reward function as well, i... more ;dly all the classical dynamic programming expected pnyolf function.\. reward function as well, in contrast to the usual randomized Markov average which depends only on the original strategy [7, 8, 11, 14, 191.
A class of empirical distributions is introduced which are based on various weighted linear combi... more A class of empirical distributions is introduced which are based on various weighted linear combinations of order statistics, and which have convergence properties the classical empirical distribution does not, or which stochastically or convexly dominate the classical empirical distribution
L evy's classical continuity theorem states that if the pointwise limit of a sequence of cha... more L evy's classical continuity theorem states that if the pointwise limit of a sequence of characteristic functions exists, then the limit function itself is a characteristic function if and only if the limit function satises a single universal limit condition (in his case, the limit at zero is one), in which case the underlying measures converge weakly to the probability measure represented by the limit function. It is the purpose of this article to give a number of direct analogs of L evy's theorem for other probability-representing functions including moment sequences, maximal moment sequences, mean-residual-life functions, Hardy-Littlewood maximal functions, and failure-rate functions. In each of these cases the single crucial condition on the limit function often relates to conservation of mass or moment, but a general theory encompassing all of these examples is still missing.
Proceedings of the American Mathematical Society, 1994
It is well known that the expected values {Mk(X)}, k > 1, of the k-maximal order statistics of an... more It is well known that the expected values {Mk(X)}, k > 1, of the k-maximal order statistics of an integrable random variable X uniquely determine the distribution of X. The main result in this paper is that if {X, }, n > 1 , is a sequence of integrable random variables with lime -00 Mk (Xn) = ak for all k > 1 , then there exists a random variable X with Mk(X) = ak for all k > 1 and Xn
Proceedings of the American Mathematical Society, 1981
Let XI, X2, . . . be independent random variables taking values in [a, b], and let T denote the s... more 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)(ba), and this bound is best possible. Probabilistically, this says that if a prophet (player with complete foresight) makes a side payment of (ba)/8 to a gambler (player using nonanticipating stop rules), the game becomes at least fair for the gambler.
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