Multitype branching processes with inhomogeneous Poisson immigration

In this paper we introduce multitype branching processes with inhomogeneous Poisson immigration, and consider in detail the critical Markov case when the local intensity r(t) of the Poisson random measure is a regularly varying function. Various multitype limit distributions (conditional and unconditional) are obtained depending on the rate at which r(t) changes with time. The asymptotic behaviour of the first and second moments, and the probability of nonextinction are investigated.

On Avakumovic’s theorem for generalized Thomas-Fermi differential equations

For the generalized Thomas-Fermi differential equation (|x?|??1x?)? = q(t)|x|??1x, it is proved that if 1 ? ? < ? and q(t) is a regularly varying function of index ? with ? > ?? ? 1, then all positive solutions that tend to zero as t ? 1 are regularly varying functions of one and the same negative index p and their asymptotic behavior at infinity is governed by the unique definite decay law. Further, an attempt is made to generalize this result to more general quasilinear differential equations of the form (p(t)|x?|??1x?)? = q(t)|x|??1x.

Moderately growing solutions of third-order differential equations with a singular nonlinearity and regularly varying coefficients

This paper is concerned with asymptotic analysis of moderately growing solutions of the third-order differential equation with singular nonlinerity((((x′)α₁*)′)α₂*)′ +q(t)x-β= 0;(A) where α1, α2and β are positive constants and σ : [α;∞)→(0;∞) is a continuous regularly varying function of index σ, α > 0 and uγ*= |u|γsgnu. An application of the theory of regular variation allows us to establish necessary and sufficient conditions for the existence of regularly varying solutions of (A) which are moderately growing and to acquire precise information about the asymptotic behavior at infinity of these solutions. The Schauder-Tychonoff fixed point technique is used.

The moment index of minima

For a random variable (RV) X its moment index κ(X) ≡ sup{κ : E(|X|κ) < ∞} lies in 0 ≤ κ (X) ≤∞; it is a critical quantity and finite for heavy-tailed RVs. The paper shows that κ (min(X, Y)) ≥ κ(X) + κ (Y) for independent non-negative RVs X and Y. For independent non-negative ‘excess' RVs Xs and Ys whose distributions are the integrated tails of X and Y, κ (X) + κ (Y) ≤ κ (min(Xs, Ys)) + 2 ≤ κ (min(X, Y)). An example shows that the inequalities can be strict, though not if the tail of the distribution of either X or Y is a regularly varying function.

The moment index of minima

For a random variable (RV) X its moment index κ(X) ≡ sup{κ : E(|X| κ ) &lt; ∞} lies in 0 ≤ κ (X) ≤∞; it is a critical quantity and finite for heavy-tailed RVs. The paper shows that κ (min(X, Y)) ≥ κ(X) + κ (Y) for independent non-negative RVs X and Y. For independent non-negative ‘excess' RVs Xs and Ys whose distributions are the integrated tails of X and Y, κ (X) + κ (Y) ≤ κ (min(Xs, Ys )) + 2 ≤ κ (min(X, Y)). An example shows that the inequalities can be strict, though not if the tail of the distribution of either X or Y is a regularly varying function.

Domain of attraction of the quasi-stationary distributions for the Ornstein-Uhlenbeck process

Let (Xt) be a one-dimensional Ornstein-Uhlenbeck process with initial density function f : ℝ+ → ℝ+, which is a regularly varying function with exponent -(1 + η), η ∊ (0,1). We prove the existence of a probability measure ν with a Lebesgue density, depending on η, such that for every A ∊ B(R+):

Domain of attraction of the quasi-stationary distributions for the Ornstein-Uhlenbeck process

Let (X t ) be a one-dimensional Ornstein-Uhlenbeck process with initial density function f : ℝ+ → ℝ+, which is a regularly varying function with exponent -(1 + η), η ∊ (0,1). We prove the existence of a probability measure ν with a Lebesgue density, depending on η, such that for every A ∊ B (R +):

On functions derived from regularly varying functions

A generalization of Karamata's theorem on integrals of regularly varying functions is proved. Using Laplace-Stieltjes transforms it is shown that any regularly varying function with exponent α (α + 1 ∉ N) is asymptotic to another regularly varying function all of whose derivations are regularly varying.

Regularly varying functions in the theory of simple branching processes

It is demonstrated for the non-critical and the explosive cases of the simple Bienaymé-Galton-Watson (B. G. W.) process (with and without immigration) that there exists a natural and intimate connection between regularly varying function theory and the asymptotic structure of the limit laws and corresponding norming constants. A similar fact had been demonstrated in connection with their invariant measures in [22]. This earlier study is complemented here by a similar analysis of the process where immigration occurs only at points of “emptiness” of the B. G. W. process.

Regularly varying functions in the theory of simple branching processes

It is demonstrated for the non-critical and the explosive cases of the simple Bienaymé-Galton-Watson (B. G. W.) process (with and without immigration) that there exists a natural and intimate connection between regularly varying function theory and the asymptotic structure of the limit laws and corresponding norming constants. A similar fact had been demonstrated in connection with their invariant measures in [22]. This earlier study is complemented here by a similar analysis of the process where immigration occurs only at points of “emptiness” of the B. G. W. process.