We show that the number generated by the q-ary integer part of an entire function of logarithmic ... more We show that the number generated by the q-ary integer part of an entire function of logarithmic order, where the function is evaluated over the natural numbers and the primes, respectively, is normal in base q. This is an extension of related results for polynomials over the real numbers established by Nakai and Shiokawa.
Dedicated to Prof. W. Narkiewicz on the occasion of his 70th birthday Abstract. All complex, pure... more Dedicated to Prof. W. Narkiewicz on the occasion of his 70th birthday Abstract. All complex, pure quartic fields with maximal orders generated by their units are determined. Furthermore, a quantitative version of the unit sum number problem is considered.
In this paper we consider an optimization problem for Ces\`aro means of bivariate functions. We a... more In this paper we consider an optimization problem for Ces\`aro means of bivariate functions. We apply methods from uniform distribution theory, calculus of variations and ideas from the theory of optimal transport.
Let K be a fleld of characteristic 0 and let (Gn(x))1 n=0 be a linear recurring sequence of degre... more Let K be a fleld of characteristic 0 and let (Gn(x))1 n=0 be a linear recurring sequence of degree d in K(x) deflned by the initial terms G0;:::;Gd¡1 2 K(x) and by the difierence equation Gn+d(x) = Ad¡1(x)Gn+d¡1(x) + ::: + A0(x)Gn(x); for n ‚ 0; with A0;:::;Ad¡1 2 K(x). Let flnally P(x) be an element of K(x). In this
It is proved that for almost all s × s-matrices A with largest eigenvalue λ(A)>1 the s... more It is proved that for almost all s × s-matrices A with largest eigenvalue λ(A)>1 the sequence of powers (A p(n)) n=1 ∞ is completely uniformly distributed modulo 1, where p(n) are different positive integers. Furthermore a constructive example for a matrix A is given such that the sequence (A n) n=1 ∞ is completely uniformly distributed.
... XC(~(t))dt -o(C)[ auf der n-dimensionalen SphEre S n formuliert. C bezeichnet eine Kugel-kapp... more ... XC(~(t))dt -o(C)[ auf der n-dimensionalen SphEre S n formuliert. C bezeichnet eine Kugel-kappe: C:{xcsn: lX-Xol<r} , SATZ 3: Es gibt ein m>O, sodaB fur ;lw-- fa stetigen Funktionen w:[O,~) ~ S n mit w(O)=w lim m ~ (m) &ilt. ... Sitzungsber. ~ster. Akad. Wiss., Math.-naturw. KI. ...
In den klassischen Modellen der kollektiven Ruintheorie wird keine Verzinsung des verfügbaren Kap... more In den klassischen Modellen der kollektiven Ruintheorie wird keine Verzinsung des verfügbaren Kapitals angenommen. Neuere Entwicklungen auf diesem Gebiet, und die allgemeine Tendenz, finanzmathematische Methoden in der Risikotheorie anzuwenden, führen zur Anwendung stochastischer Zinssätze in der Ruintheorie. Außerdem diskutieren wir die Simulation des Problems mit hybrid Monte Carlo Verfahren.
H. U. Gerber [Scand. Actuarial J. 1981, 105-115 (1981; Zbl 0455.62086)] investigates the followin... more H. U. Gerber [Scand. Actuarial J. 1981, 105-115 (1981; Zbl 0455.62086)] investigates the following actuarial model: The premium flow is continuous with a constant intensity c. The claim number variable is Poisson-distributed with parameter λ; the claim size distribution is given by F(y)=1-e -y . If the capital reaches a barrier b+at, it stays on the barrier up to the next claim. Guided by D. C. M. Dickson and J. R. Gray [ibid. 1984, 174-186 (1984; Zbl 0557.62086)], who consider the solution of the model in case of a horizontal barrier, the authors compute the survival probability and the expected value of the dividends for mixed-exponential- and gamma-distributed claim sizes, using the software package MAPLE. Furthermore they demonstrate how this model can be simulated and how a solution can be found by an integration-method. All these procedures are compared with the help of detailed tests.
A model in collective ruin theory is developed to cope with conditions typical for alternative re... more A model in collective ruin theory is developed to cope with conditions typical for alternative reinsurance over the capital markets, where contracts are tailored to easily accessible information and therefore often do not provide perfect covers for the risk that are securitized that way. Examples include the dependency of payments on claim numbers or index states. The dynamics of both is modeled by a n-state Markov chain, each state with possibly different claim frequencies and claim distributions. A fixed non-negative payment will accompany each state change which may be triggered externally or by a claim. The model for the non-ruin probability leads to a system of retarded integro-differential equations which can be solved analytically.
ABSTRACT The main part of this paper deals with limitibility properties of weakly dependent unifo... more ABSTRACT The main part of this paper deals with limitibility properties of weakly dependent uniformly bounded random variables with respect to weighted means and a uniform weighted limitation method. In the next part a weighted Le-Roy-limitation method is introduced and discussed. For example it is shown that every Mp-limitable sequence is limitable with respect to the weighted Le-Roy method.
ABSTRACT By a classical principle of probability theory, sufficiently thin subsequences of genera... more ABSTRACT By a classical principle of probability theory, sufficiently thin subsequences of general sequences of random variables behave like i.i.d. sequences. This observation not only explains the remarkable properties of lacunary trigonometric series, but also provides a powerful tool in many areas of analysis, such the theory of orthogonal series and Banach space theory. In contrast to i.i.d. sequences, however, the probabilistic structure of lacunary sequences is not permutation-invariant and the analytic properties of such sequences can change after rearrangement. In a previous paper we showed that permutation-invariance of subsequences of the trigonometric system and related function systems is connected with Diophantine properties of the index sequence. In this paper we will study permutation-invariance of subsequences of general r.v. sequences.
We show that the number generated by the q-ary integer part of an entire function of logarithmic ... more We show that the number generated by the q-ary integer part of an entire function of logarithmic order, where the function is evaluated over the natural numbers and the primes, respectively, is normal in base q. This is an extension of related results for polynomials over the real numbers established by Nakai and Shiokawa.
Dedicated to Prof. W. Narkiewicz on the occasion of his 70th birthday Abstract. All complex, pure... more Dedicated to Prof. W. Narkiewicz on the occasion of his 70th birthday Abstract. All complex, pure quartic fields with maximal orders generated by their units are determined. Furthermore, a quantitative version of the unit sum number problem is considered.
In this paper we consider an optimization problem for Ces\`aro means of bivariate functions. We a... more In this paper we consider an optimization problem for Ces\`aro means of bivariate functions. We apply methods from uniform distribution theory, calculus of variations and ideas from the theory of optimal transport.
Let K be a fleld of characteristic 0 and let (Gn(x))1 n=0 be a linear recurring sequence of degre... more Let K be a fleld of characteristic 0 and let (Gn(x))1 n=0 be a linear recurring sequence of degree d in K(x) deflned by the initial terms G0;:::;Gd¡1 2 K(x) and by the difierence equation Gn+d(x) = Ad¡1(x)Gn+d¡1(x) + ::: + A0(x)Gn(x); for n ‚ 0; with A0;:::;Ad¡1 2 K(x). Let flnally P(x) be an element of K(x). In this
It is proved that for almost all s × s-matrices A with largest eigenvalue λ(A)>1 the s... more It is proved that for almost all s × s-matrices A with largest eigenvalue λ(A)>1 the sequence of powers (A p(n)) n=1 ∞ is completely uniformly distributed modulo 1, where p(n) are different positive integers. Furthermore a constructive example for a matrix A is given such that the sequence (A n) n=1 ∞ is completely uniformly distributed.
... XC(~(t))dt -o(C)[ auf der n-dimensionalen SphEre S n formuliert. C bezeichnet eine Kugel-kapp... more ... XC(~(t))dt -o(C)[ auf der n-dimensionalen SphEre S n formuliert. C bezeichnet eine Kugel-kappe: C:{xcsn: lX-Xol<r} , SATZ 3: Es gibt ein m>O, sodaB fur ;lw-- fa stetigen Funktionen w:[O,~) ~ S n mit w(O)=w lim m ~ (m) &ilt. ... Sitzungsber. ~ster. Akad. Wiss., Math.-naturw. KI. ...
In den klassischen Modellen der kollektiven Ruintheorie wird keine Verzinsung des verfügbaren Kap... more In den klassischen Modellen der kollektiven Ruintheorie wird keine Verzinsung des verfügbaren Kapitals angenommen. Neuere Entwicklungen auf diesem Gebiet, und die allgemeine Tendenz, finanzmathematische Methoden in der Risikotheorie anzuwenden, führen zur Anwendung stochastischer Zinssätze in der Ruintheorie. Außerdem diskutieren wir die Simulation des Problems mit hybrid Monte Carlo Verfahren.
H. U. Gerber [Scand. Actuarial J. 1981, 105-115 (1981; Zbl 0455.62086)] investigates the followin... more H. U. Gerber [Scand. Actuarial J. 1981, 105-115 (1981; Zbl 0455.62086)] investigates the following actuarial model: The premium flow is continuous with a constant intensity c. The claim number variable is Poisson-distributed with parameter λ; the claim size distribution is given by F(y)=1-e -y . If the capital reaches a barrier b+at, it stays on the barrier up to the next claim. Guided by D. C. M. Dickson and J. R. Gray [ibid. 1984, 174-186 (1984; Zbl 0557.62086)], who consider the solution of the model in case of a horizontal barrier, the authors compute the survival probability and the expected value of the dividends for mixed-exponential- and gamma-distributed claim sizes, using the software package MAPLE. Furthermore they demonstrate how this model can be simulated and how a solution can be found by an integration-method. All these procedures are compared with the help of detailed tests.
A model in collective ruin theory is developed to cope with conditions typical for alternative re... more A model in collective ruin theory is developed to cope with conditions typical for alternative reinsurance over the capital markets, where contracts are tailored to easily accessible information and therefore often do not provide perfect covers for the risk that are securitized that way. Examples include the dependency of payments on claim numbers or index states. The dynamics of both is modeled by a n-state Markov chain, each state with possibly different claim frequencies and claim distributions. A fixed non-negative payment will accompany each state change which may be triggered externally or by a claim. The model for the non-ruin probability leads to a system of retarded integro-differential equations which can be solved analytically.
ABSTRACT The main part of this paper deals with limitibility properties of weakly dependent unifo... more ABSTRACT The main part of this paper deals with limitibility properties of weakly dependent uniformly bounded random variables with respect to weighted means and a uniform weighted limitation method. In the next part a weighted Le-Roy-limitation method is introduced and discussed. For example it is shown that every Mp-limitable sequence is limitable with respect to the weighted Le-Roy method.
ABSTRACT By a classical principle of probability theory, sufficiently thin subsequences of genera... more ABSTRACT By a classical principle of probability theory, sufficiently thin subsequences of general sequences of random variables behave like i.i.d. sequences. This observation not only explains the remarkable properties of lacunary trigonometric series, but also provides a powerful tool in many areas of analysis, such the theory of orthogonal series and Banach space theory. In contrast to i.i.d. sequences, however, the probabilistic structure of lacunary sequences is not permutation-invariant and the analytic properties of such sequences can change after rearrangement. In a previous paper we showed that permutation-invariance of subsequences of the trigonometric system and related function systems is connected with Diophantine properties of the index sequence. In this paper we will study permutation-invariance of subsequences of general r.v. sequences.
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