Infinitely divisible distributions arising from first crossing times and related results

Abstract

Results of Jain and Khan (1979) and Khan and Jain (1978) on the time to first emptiness of a reservoir are generalized to include the case of defective random variables where the mass at ∞ can be positive. The assumption of an underlying exponential family is not needed — the general condition is infinite divisibility and closure under convolutions. The support of the distributions can be nonnegative reals or nonnegative integers. Examples are given to illustrate the general theory and show the bivariate extension.

Appendix

1.1 A.1 Natural exponential family; an example

Let f ₀(z) be a density (say, with respect to Lebesgue or counting measure) with cumulant generating function ψ(t) over a range t ∈ Θ where the moment generating function e ^ψ is finite. Then, as in Morris (1982), the exponential family

$$ e^{\theta z-\psi(\theta)} f_0(z), \quad \theta\in \Theta, $$

can be derived. Its mean and variance are ψ′(θ) and ψ′′(θ) respectively. If f ₀(z) = f ₀(z;λ) is infinitely divisible with convolution parameter λ, then the cumulant generating function has the form of ψ(t) = ψ _λ (t) = λψ ₁(t), where ψ ₁(t) is the cumulant generating function of f ₀(z;1). The exponential family with “natural parameter” θ and convolution parameter λ > 0 is

$$ e^{\theta z-\lambda\psi_1(\theta)} f_0(z;\lambda), \quad \theta\in \Theta, \quad \lambda>0. $$

(A.1)

For non-Poisson families, a non-convolution parameter appears in f ₀ and it combines with θ in (A.1). If f ₀(z;λ) is Poisson(λ), then (A.1) is Poisson(λe ^θ). From its definition, it follows that (A.1) has cumulant generating function λ[ψ ₁(θ + t) − ψ ₁(t)] for t such that θ ∈ Θ, θ + t ∈ Θ; furthermore its mean and variance are respectively λψ′₁(θ) and λψ′′₁(θ) for θ ∈ Θ.

This is illustrated for the negative binomial distribution NB(λ,ξ) with 0 < ξ < 1 fixed and q = 1 − ξ. The probability mass function is

$$ f_0(z;\lambda)=\frac{\Gamma(\lambda+z)}{\Gamma(\lambda)\,z!} \xi^\lambda q^z, \quad z=0,1,2,\ldots, $$

and its cumulant generating function is $\psi_\lambda(t)= \lambda[\log \xi-\log(1-qe^t)]$ and Θ = {t: qe ^t < 1}. The natural exponential family in (A.1) is

$$ e^{\theta z-\lambda[\log \xi-\log(1-qe^\theta)]} \frac{\Gamma(\lambda+z)}{\Gamma(\lambda)\,z!} \xi^\lambda q^z = \frac{\Gamma(\lambda+z)}{\Gamma(\lambda)\,z!} (1-qe^\theta)^\lambda (qe^\theta)^x. $$

(A.2)

That is, this is the NB(λ,1 − qe ^θ) distribution and the natural exponential family generated from NB(λ,ξ) is NB with a different parametrization of the non-convolution parameter. For the parametrization of (A.2), the mean is μ(θ) = λqe ^θ/(1 − qe ^θ), the variance is V(θ) = λqe ^θ/(1 − qe ^θ)² and V(θ) = μ(θ) + μ ²(θ)/λ for all θ ∈ Θ.

1.2 A.2 Results on derivatives of univariate and bivariate Laplace transforms

For the first two results, let Y be a nonnegative random variable with finite mean μ _Y and LT L _Y (s).

Result A1. |L′ _Y (s)/L _Y (s)| ≤ μ _Y for all s ≥ 0.

Proof

Note that $|L'_Y(s)|/L_Y(s)=\int_0^\infty ye^{-sy}dF_Y(y)/L_Y(s)$. Consider the family of densities with $g(y;s)=e^{-sy}f_Y(y)/L_Y(s)$ on Lebesgue or counting measure. As s increases, the density g(·;s) puts less weight on large values of y. Hence g(·;s) is stochastic decreasing as s increases (from page 261 of Marshall and Olkin 2007, this family is decreasing in the stronger likelihood ratio order). Hence the expected value of Y ^* with density g(·;s) is decreasing in s. The expected value for s = 0 is μ _Y and hence this is a bound.

Result B1. lim _s→ ∞ |L′ _Y (s)/L _Y (s)| = 0.

Proof

For densities g(·;s) defined above, the family converges in distribution to the degenerate random variable 0 as s→ ∞.

The above results can be extended to bivariate LTs with similar proofs. Let (Y ₁,Y ₂) be a nonnegative random pair with finite means $\mu_{Y_1},\mu_{Y_2}$ and LT $L_{Y_1,Y_2}(s_1,s_2)$.

Result A2.

$$ \left|\frac{\partial L_{Y_1,Y_2}(s_1,s_2)}{\partial s_j}\right|\left /\vphantom{\frac{\partial L_{Y_1,Y_2}(s_1,s_2)}{\partial s_j}}\right. L_{Y_1,Y_2}(s_1,s_2)\le \mu_{Y_j}, $$

j = 1,2, for all s ₁,s ₂ ≥ 0.

Proof

$$ \frac{|\frac{\partial L_{Y_1,Y_2}(s_1,s_2)}{ \partial s_j}| }{ L_{Y_1,Y_2}(s_1,s_2)} =\frac{\int_0^\infty y_je^{-s_1y_1-s_2y_2}dF_{Y_1,Y_2}(y_1,y_2) }{ L_{Y_1,Y_2}(s_1,s_2)}. $$

Consider the family of densities with

$$ g(y_1,y_2;s_1,s_2)=e^{-s_1y_1-s_2y_2}f_{Y_1,Y_2}(y_1,y_2) /L_{Y_1,Y_2}(s_1,s_2) $$

on Lebesgue or counting measure. As s ₁,s ₂ increase, the density g(·;s ₁,s ₂) puts less weight on large values of y ₁,y ₂. Hence g(·;s ₁,s ₂) is stochastic decreasing as s ₁,s ₂ increase. Hence the expected values of random variables $Y_1^*,Y_2^*$ with density g(·;s ₁,s ₂) are decreasing in s ₁,s ₂. The expected value for s ₁ = s ₂ = 0 are $\mu_{Y_1},\mu_{Y_2}$ and hence these are bounds.

Result B2.

$$ \lim\limits_{s\to\infty} \left|\frac{\partial L_{Y_1,Y_2}(sr_1,sr_2)}{\partial s_j}\left/\vphantom{\frac{\partial L_{Y_1,Y_2}(sr_1,sr_2)}{\partial s_j}}\right. L_{Y_1,Y_2}(sr_1,sr_2)\right|=0. $$

Proof

For densities g(·;s ₁,s ₂) defined above, the family converges in distribution to the degenerate random vector (0,0) as s→ ∞ with s ₁ = sr ₁ and s ₂ = sr ₂.