Asymptotic normality in a coupon collector's problem

Summary Let {a _s , s=1, 2, ..., N} be a set of reals and {p _s , s=1, 2, ..., N} be a set of probabilities, i.e. p _s0 and p ₁+p ₂+...+p _N =1. Let I ₁ I ₂,... be independent random variables, all with the distribution P(I=s)=p _s , s=1, 2, ..., N. Put U _v =l if I _v {I ₁, I ₂, ..., I _v –1} and U _v =0 otherwise, v=1, 2, .... The random variable Z _n = is called the bonus sum after ncoupons for a coupon collector in the situation {(p _s , a _s ), s=1, 2, ..., N}.Consider a sequence {(p _ks , a _ks ), s=l, 2, ..., N _k }, k=1, 2, ..., of collector situations, and let {Z _n ^(k) , n=1, 2, ...}, k=1, 2, ..., be the corresponding sequence of bonus sum variables. Let d be an arbitrary natural number and let , k=1, 2, ..., where 1 n _k ⁽¹⁾<n _k ⁽²⁾<< n _k ^(d) .We assume that N ^(k) t8 and that .It is shown that the random vector V ^(k) is, under general conditions, asymptotically (as kt8) normally distributed. An asymptotic expression for the covariance matrix of V ^(k) is derived.Research supported in part at Stanford University, Stanford, California under contract N0014-67-A-0112-0015.     

On Hardy inequalities with a remainder term

In this paper we study some improvements of the classical Hardy inequality. We add to the right hand side of the inequality a term that depends on some Lorentz norms of u or of its gradient and we find the best values of the constants for remaining terms. In both cases we show that the problem of finding the optimal value of the constant can be reduced to a spherically symmetric situation. This result is new when the right hand side is a Lorentz norm of the gradient.     

On the remainder term in theN-dimensional Euler Maclaurin expansion

The form of the remainder term in theN-dimensional Euler Maclaurin expansion is investigated. A concise formalism is developed for handling expressions which are lengthy to state using conventional notation. Conditions under which an integral representation involving only derivatives of the same total order with conventional kernel functions exists for the remainder term are derived.Work performed in part under the auspices of the U. S. Atomic Energy Commission.Part of this work was carried out by both authors at the University of N.S.W., Kensington, N.S.W., Australia.     

On the remainder term in the circle problem in an arithmetic progression

In this paper we improve the estimate for the remainder term in the asymptotic formula concerning the circle problem in an arithmetic progression.     

Improvement of remainder term for the divisors problem

New bounds are obtained for the remainder term in the divisors problem.Translated from Matematicheskie Zametki, Vol. 6, No. 5, pp. 545–554, November, 1969.     

The remainder term in quadrature formulae

Summary It is shown that the remainder term of any quadrature formula has an asymptotic expansion in terms of the step size; the occurrence of remainder terms of formc f ⁽ⁿ⁺¹⁾ () is discussed.     

Estimate of the remainder term in Tate's formula

An estimate of the remainder term in Tate's formula is deduced and a property of finite order points of elliptic curves is pointed out.Translated from Matematicheskie Zametki, Vol. 3, No. 3, pp. 271–278, March, 1968.     

On the remainder term in the Weyl law for cofinite Kleinian groups with finite dimensional unitary representation

We obtain a new bound for the remainder term in the Weyl law for general cofinite Kleinian groups with finite dimensional unitary representation by applying a Tauberian theorem for the Laplace transform and a suitable truncation argument.     

Evaluation of the remainder term for the Thiele interpolation continued fraction

We present an estimate of the remainder term for the Thiele interpolation continued fraction. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 11, pp. 1548–1554, November, 2008.     

Oscillations of the remainder term related to the Euler totient function

We split the remainder term in the asymptotic formula for the mean of the Euler phi function into two summands called the arithmetic and the analytic part respectively. We show that the arithmetic part can be studied with a mild use of the complex analytic tools, whereas the study of the analytic part heavily depends on the properties of the Riemann zeta function and on the distribution of its non-trivial zeros in particular.