D. V. Gusak

On extrema of integer-valued processes, defined by sums of a random number of addenda

On extrema of integer-valued processes, defined by sums of a random number of addenda D. V. GUSAK and A. M. ROZUMENKO Institute of Mathematics of the Ukrainian Academy of Sciences, Kiev, Ukraine Received for ROSE 19 January 1995 Abstract — For a lattice random walk {£(^))^ ^ 0} denned on the superposition of two renewal processes, the distributions of extremal values: ί ( 0 = sup ( 0<u<t are studied. The question of the distributions of pairs {i(<uw}, u(*uw}. mrw), {rcuw is also investigated.

Oscillating processes with independent increments and nondegenerate Wiener component

For an oscillating process ζz(t) (ζz(0)=2,t⩾0), which is defined with the help of two homogeneous processes ξ1(t) and ξ2(t) with independent increments and nondegemerate Wiener components, under certain restrictions we establish a relation of the form and find the characteristic function of the ergodic distribution of the process considered.

The cumulant representation of the Lundberg root in the case of semicontinuous processes

For the case of homogeneous processes ξ(t), ξ(0) = 0, t ≥ 0, with independent increments and negative jumps, it is proved in A. V. Skorokhod, Random Processes with Independent Increments, Nauka, Moscow, 1964 that the functional τ(x) = inf {t ≥ 0 : ξ(t) > x} , x ≥ 0, is a nondecreasing process with independent increments with respect to x, and its moment generating function is expressed via the cumulant that satisfies the corresponding Lundberg equation. The corresponding representations of this cumulant are specified and its Lévy characteristics (namely, γ and Lévy’s integral measure N(x)) are evaluated by using some of the results of the author’s work of 2007 for the processes under consideration.

ON THE FIRST RUIN MOMENT FOR A MODIFIED RISK PROCESS WITH IMMEDIATE REFLECTION

For a modified risk process with immediate reflection downward, we establish relations for an integral transformation of its characteristic function and the corresponding transformation of the limit distribution of the considered process under ergodicity conditions. The distribution is obtained for the first ruin moment of the introduced risk process.

Prelimit and limit generalizations of the Pollaczek–Khinchin formula

The moment generating function of the nondegenerate distribution of the maximum ξ+ = sup0≤t<∞ ξ(t) of a compound Poisson process ξ(t) = at+ S(t), a < 0, S(t) = ∑ k≤ν(t) ξk, ξk > 0, where ν(t) is a simple Poisson process with intensity λ > 0, is determined via the well-known Pollaczek–Khinchin formula if m = Eξ(1) < 0. We obtain a prelimit generalization of this formula that determines the Laplace– Carson transform of the moment generating function of the maximum ξ+(t) = sup0≤t′≤t ξ(t ′), 0 < t < ∞, and the moment generating function of ξ+ = ξ+(∞) under the assumption that m < 0 for homogeneous processes ξ(t) with independent increments and of bounded variation. Relationships of a different type between characteristic functions of ξ(θs) (P{θs > t} = e−st, s, t > 0) and of ξ+ are also obtained by using earlier results presented by the author. Introduction Let ξ(t) be a lower continuous compound Poisson process whose characteristic function is defined by the cumulant ψ(α), that is, E e = e, t ≥ 0, ψ(α) = iαa+ λ(φ(α)− 1), a < 0, λ > 0, φ(α) = E ek , F (x) = P{ξk < x}, x ≥ 0, F (x) = 1− F (x). If m = E ξ(1) = a+ λμ1 < 0, μ1 = E ξk, F (0) = 0, then the moment generating function of ξ = sup0≤t<∞ ξ(t) is determined via the Pollaczek–Khinchin formula (1) E e−zξ + = p+ 1− q+F̃ (z)/μ1 , p+ = P{ξ = 0} = 1− q+,

Exit time functionals for integer-valued Poisson processes

The joint distribution of all exit time functionals is studied in this paper for a fixed level x and integer-valued compound Poisson processes. An exact formula for the distributions of these functionals is obtained in the case of semicontinuous processes. Limit relations are obtained for the distributions of the exit time functionals for x = 0 or as x→∞. The distribution of exit time functionals is studied in [1]–[3] for homogeneous processes with independent increments and with jumps whose distribution function is continuous. The same problem is studied in [4, 5] for integer-valued processes ξ(t) defined on finite Markov chains. Moreover, limit distributions as x → ∞ (x is an integer) are given in [4, 5] under the condition that m1 = E ξ(1) ≥ 0. We obtain an exact formula for the distribution of exit time functionals in the case of semicontinuous compound Poisson processes for both casesm1 ≥ 0 andm1 < 0. As a corollary of these relations we obtain an assertion on the distribution of extreme values of a semicontinuous integer-valued Poisson process. In particular, we obtain a discrete analogue of the Pollaczek–Khintchine formula for the distribution of the global maximum in the case of m1 < 0. Consider an integer-valued Poisson process ξ(t), t ≥ 0, such that ξ(0) = 0 and the moment generating function and cumulant are given by (1) pt(u) = Eu = e, |u| = 1, and k(u) = λ ∑ k 6=0 ( u − 1 ) pk = λ(p(u)− 1), |u| = 1, λ > 0,

On the Creative Contribution of V. S. Korolyuk to the Development of Probability Theory

We present a brief survey of the main results obtained by V. S. Korolyuk in probability theory and mathematical statistics.

Ruin problem for an inhomogeneous semicontinuous integer-valued process

For a process ξ(t = ξ1(t)+χ(t), t≥0, ξ(0) = 0, inhomogeneous with respect to time, we investigate the ruin problem associated with the corresponding random walk in a finite interval, (here, ξ1 (t) is a homogeneous Poisson process with positive integer-valued jumps and χ(t) is an inhomogeneous lower-semicontinuous process with integer-valued jumps ξn≥-1).

Limit behavior of the distribution of the ruin moment of a modified risk process

For a modified risk process with instantaneous reflection at the pointB>0 under which the precess considered

Basic identities for additive continuously distributed sequences

For an additive sequence ξ(n), we establish basic factorization identities and express the distributions of limiting Junctionals (extremum values of ξ(n), the time and value of the first jump over a fixed level, etc.) in terms of the components of factorization.