David Gilat

On the distribution of maxima of martingales

Partial order the set of distributions on the real line by v 1, the classical moment inequality that the Lp norm of Ai (and of ,u*) is at most p/(p - 1) times the 4L norm of ,u is shown to be sharp.

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, extending classical theorems; of Hoeffding and of Helmers for lid 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.

On the structure of 1-dependent Markov chains

Any stationary 1-dependent Markov chain with up to four states is a 2-block factor of independent, identically distributed random variables. There is a stationary 1-dependent Markov chain with five states which is not, even though every 1-dependent renewal process is a 2-block factor.

On the ratio of the expected maximum of a martingale and theLp-norm of its last term

For eachp>1, the supremum,S, of the absolute value of a martingale terminating at a random variableX inLp, satisfiesES≦(Γ(q))1/q‖X‖p (q=p(p-1)-1).The maximum,M, of a mean-zero martingale which starts at zero and terminates atX, satisfiesES≦(Γ(q))1/q‖X‖p (q=p(p-1)-1), whereσq is the unique solution of the equationt = ‖Z −t ‖q for an exponentially distributed random variableZ with mean 1.σp has other characterizations and satisfies limp‖q− 1σq =c withc determined bycec+1 = 1. Equalities in (1) and (2) are attainable by appropriate martingales which can be realized as stopped segments of Brownian motion. A presumably new property of the exponential distribution is obtained en route to inequality (2).

On the expected diameter of an L2-bounded martingale

Dedicated to the memory of Gideon Schwarz (1933-2007) It is shown that the ratio between the expected diameter of an L2-bounded martingale and the standard deviation of its last term cannot exceed p 3. Moreover, a one-parameter family of stopping times on standard Brownian Motion is exhibited, for which the p 3 upper bound is attained. These stopping times, one for each cost-rate c, are optimal when the payoff for stopping at time t is the diameter D(t) obtained up to time t minus the hitherto accumulated cost ct. A quantity related to diameter, maximal drawdown (or rise), is introduced and its expectation is shown to be bounded by p 2 times the standard deviation of the last term of the martingale. These results complement the Dubins & Schwarz respective bounds 1 and p 2 for the ratios between the expected maximum and maximal absolute value of the martingale and the standard deviation of its last term. Dynamic programming (gambling theory) methods are used for the proof of optimality.

Strongly-consistent, distribution-free confidence intervals for quantiles

Strongly-consistent, distribution-free confidence intervals are derived to estimate the fixed quantiles of an arbitrary unknown distribution, based on order statistics of an iid sequence from that distribution. This new method, unlike classical estimates, works for totally arbitrary (including discontinuous) distributions, and is based on recent one-sided strong laws of large numbers.

Gauss's Lemma and the Irrationality of Roots, Revisited

Summary An idea of T. Estermann (1975) for demonstrating the irrationality of √2 is extended to obtain a conceptually simple proof of Gausss Lemma, according to which real roots of monic polynomials with integer coefficients are either integers or irrational. The standard proof of the lemma is also reviewed.