For a set A of positive integers and any positive integer n, let $R_{1}(A, n)$ , $R_{2}(A,n)$ and $R_{3}(A,n)$ denote the number of solutions of $a+a^{\prime }=n$ with $a, a^{\prime }\in A$ and the additional restriction that $a<a^{\prime }$ for $R_{2}$ and $a\leq a^{\prime }$ for $R_{3}$ . We consider Problem 6 of Erdős et al. [‘On additive properties of general sequences’, Discrete Math. 136 (1994), 75–99] about locally small and locally large values of $R_{1}, R_{2}$ and $R_{3}$ .

For a given set $S\subset \mathbb{N}$, $R_{S}(n)$ is the number of solutions of the equation $n=s+s^{\prime },s<s^{\prime },s,s^{\prime }\in S$. Suppose that $m$ and $r$ are integers with $m>r\geq 0$ and that $A$ and $B$ are sets with $A\cup B=\mathbb{N}$ and $A\cap B=\{r+mk:k\in \mathbb{N}\}$. We prove that if $R_{A}(n)=R_{B}(n)$ for all positive integers $n$, then there exists an integer $l\geq 1$ such that $r=2^{2l}-1$ and $m=2^{2l+1}-1$. This solves a problem of Chen and Lev [‘Integer sets with identical representation functions’, Integers16 (2016), A36] under the condition $m>r$.

We derive single integral representations for the exact distribution of the sum of independent generalized Pareto random variables. The integrands involve the incomplete and complementary incomplete gamma functions. Applications to insurance and catastrophe bonds are described.

The explicit form for the characteristic function of a stable distribution on the line is derived analytically by solving the associated functional equation and applying the theory of regular variation, without appeal to the general Lévy‒Khintchine integral representation of infinitely divisible distributions.

The problem of pricing arithmetic Asian options is nontrivial, and has attracted much interest over the last two decades. This paper provides a method for calculating bounds on option prices and approximations to option deltas in a market where the underlying asset follows a geometric Lévy process. The core idea is to find a highly correlated, yet more tractable proxy to the event that the option finishes in-the-money. The paper provides a means for calculating the joint characteristic function of the underlying asset and proxy processes, and relies on Fourier methods to compute prices and deltas. Numerical studies show that the lower bound provides accurate approximations to prices and deltas, while the upper bound provides good though less accurate results.

We study a family of distributions that satisfy the stability-under-addition property, provided that the number ν of random variables in a sum is also a random variable. We call the corresponding property ν-stability and investigate the situation when the semigroup generated by the generating function of ν is commutative. Using results from the theory of iterations of analytic functions, we describe ν-stable distributions generated by summations with rational generating functions. A new case in this class of distributions arises when generating functions are linked with Chebyshev polynomials. The analogue of normal distribution corresponds to the hyperbolic secant distribution.

We use a change-of-variable formula in the framework of functions of bounded variation to derive an explicit formula for the Fourier transform of the level crossing function of shot noise processes with jumps. We illustrate the result in some examples and give some applications. In particular, it allows us to study the asymptotic behavior of the mean number of level crossings as the intensity of the Poisson point process of the shot noise process goes to infinity.

We study when characteristic and Hölder continuous functions are traces of Sobolev functions on doubling metric measure spaces. We provide analytic and geometric conditions sufficient for extending characteristic and Hölder continuous functions into globally defined Sobolev functions.

We consider the random motion of a particle that moves with constant finite speed in the space ℝ4 and, at Poisson-distributed times, changes its direction with uniform law on the unit four-sphere. For the particle's position, X(t) = (X1(t), X2(t), X3(t), X4(t)), t > 0, we obtain the explicit forms of the conditional characteristic functions and conditional distributions when the number of changes of directions is fixed. From this we derive the explicit probability law, f(x, t), x ∈ ℝ4, t ≥ 0, of X(t). We also show that, under the Kac condition on the speed of the motion and the intensity of the switching Poisson process, the density, p(x,t), of the absolutely continuous component of f(x,t) tends to the transition density of the four-dimensional Brownian motion with zero drift and infinitesimal variance σ2 = ½.

Let $w(z)$ be an arbitrary transcendental solution of the fourth (respectively, second) Painlevé equation. Concerning the frequency of poles in $|z|\le r$, it is shown that $n(r,w)\gg r^2$ (respectively, $n(r,w)\gg r^{3/2}$), from which the growth estimate $T(r,w)\gg r^2$ (respectively, $T(r,w)\gg r^{3/2}$) immediately follows.

AMS 2000 Mathematics subject classification: Primary 34M55; 34M10

This paper has as its main theme the fitting in practice of the variance-gamma distribution, which allows for skewness, by moment methods. This fitting procedure allows for possible dependence of increments in log returns, while retaining their stationarity. It is intended as a step in a partial synthesis of some ideas of Madan, Carr and Chang (1998) and of Heyde (1999). Standard estimation and hypothesis-testing theory depends on a large sample of observations which are independently as well as identically distributed and consequently may give inappropriate conclusions in the presence of dependence.

Let ζ be a Markov chain on a finite state space D, f a function from D to ℝd, and Sn = ∑k=1nf(ζk). We prove an invariance theorem for S and derive an explicit expression of the limit covariance matrix. We give its exact value for p-reinforced random walks on ℤ2 with p = 1, 2, 3.

This paper establishes a compound Poisson limit theorem for the sum of a sequence of multi-state Markov chains. Our theorem generalizes an earlier one by Koopman for the two-state Markov chain. Moreover, a similar approach is used to derive a limit theorem for the sum of the k th-order two-state Markov chain.

In this paper we provide further results on the general d-dimensional correlated random walk. In particular, we prove that the n-step characteristic function of any correlated random walk satisfies a recurrence formula which enables both it and the total characteristic function to be obtained. Some examples are then considered.

A Markovian stochastic model for a system subject to random shocks is introduced. It is assumed that the shock arriving according to a Poisson process decreases the state of the system by a random amount. It is further assumed that the system is repaired by a repairman arriving according to another Poisson process if the state when he arrives is below a threshold α. Explicit expressions are deduced for the characteristic function of the distribution function of X(t), the state of the system at time t, and for the distribution function of X(t), if . The stationary case is also discussed.

We define an iterative interpolation process for data spread over a closed discrete subgroup of the Euclidean space. We describe the main algebraic properties of this process. This interpolation process, under very weak assumptions, is always convergent in the sense of Schwartz distributions. We find also a convenient necessary and sufficient condition for continuity of each interpolation function of a given iterative interpolation process.

A two-dimensional lattice random walk is studied in which, at each stage, the direction of the next step is correlated to that of the previous step. An exact expression is obtained from the characteristic function for the dispersion matrix of the n-step transition probabilities, and the limiting diffusion equation is derived.

The busy period of the GI/D/∞ queue is determined as the time when at least one customer is served. Let v be the number of customers served during this period. The busy period of order n is defined as a busy period for which v ≧ n. In this paper we derive the exact distributions, integral equations, characteristic functions and all moments of those periods. Finally, some properties of the idle periods of order n are established.