A complete comparison is made between the value V(X1,..., Xn) = sup{EXt: t is a stop rule for X1,...,Xn} and E(maxj = nXj) for all uniformly bounded sequences of i.i.d. random variables X1, ..., Xn. Specifically, the set of ordered pairs {(x,y): X = V(X1, ..., Xn) and Y = E(maxj=nXj) for some...

Consider evolution of density of a mass or a population, geographically situated in a compact region of space, assuming random creation-annihilation and migration, or dispersion of mass, so the evolution is a random measure. When the creation-annihilation and dispersion are diffusions the...

Let M be a 4N-integrable, real-valued continuous N-parameter strong martingale. By extending Itô-type formulas for M to a function whose 2Nth derivative is Dirac's [delta]-distribution, Tanaka-type formulas for M are obtained. They represent local time of M with respect to occupation time...

Exact comparisons are made relating EY0p, EYn-1p, and E(maxj<=n-1 Yjp), valid for all martingales Y0,...,Yn-1, for each p >= 1. Specifically, for p 1, the set of ordered triples {(x, y, z) : X = EY0p, Y = E Yn-1p, and Z = E(maxj<=n-1 Yjp) for some martingale Y0,...,Yn-1} is precisely the set {(x, y, z) : 0<=x<=y<=z<=[Psi]n,p(x, y)}, where [Psi]n,p(x, y) = x[psi]n,p(y/x) if x > 0, and = an-1,py if x = 0; here [psi]n,p is a specific recursively defined function. The result yields families of sharp...</=n-1></=n-1>

Operator-stable laws and operator-semistable laws (introduced as limit distributions by M. Sharpe and R. Jajte, respectively) are characterized by decomposability properties. Disintegration of their corresponding Lévy measures requires appropriate cross sections. Furthermore in both situations...

We introduce a new condition for {Y[tau]n} to have the same asymptotic distribution that {Yn} has, where {Yn} is a sequence of random elements of a metric space (S, d) and {[tau]n} is a sequence of random indices. The condition on {Yn} is that maxi[set membership, variant]Dnd(Yi, Yan)--p0 as n...

Let F be a Banach space with a sufficiently smooth norm. Let (Xi)i<=n be a sequence in LF2, and T be a Gaussian random variable T which has the same covariance as X = [Sigma]i<=nXi. Assume that there exists a constant G such that for s, [delta]>=0, we have P(s[less-than-or-equals, slant]||T||[less-than-or-equals, slant]s+[delta])[less-than-or-equals, slant]G[delta]. (*) We then give explicit bounds of [Delta](X) = supiP(X=t)-P(T=t) in terms of truncated moments of the...</=n>

This paper is devoted to tests for uniformity based on sum-functions of m-spacings, where m diverges to infinity as the sample size, n, increases. It is shown that if m diverges at a slower rate than n1/2 then the commonly used sum-function will detect alternatives distant (mn)-1/4 from the...

For the purpose of comparing different nonparametric density estimators, Wegman (J. Statist. Comput. Simulation 1 225-245) introduced an empirical error criterion. In a recent paper by Hall (Stochastic Process. Appl. 13 11-25) it is shown that this empirical error criterion converges to the mean...

A sequence of independent, identically distributed random vectors X1, X2, X3,... is said to belong to the domain of attraction of a random vector Y is there exist linear operators An and constant vectors bn such that An(X1,..., Xn)+bn converges in distribution to Y. We present a simple,...