Convergence of extreme values of Poisson point processes at small times

Abstract

We study the behaviour of large values of extremal processes at small times, obtaining an analogue of the Fisher-Tippet-Gnedenko Theorem. Thus, necessary and sufficient conditions for local convergence of such maxima, linearly normalised, to the Fréchet or Gumbel distributions, are established. Weibull distributions are not possible limits in this situation. Moreover, assuming second order regular variation, we prove local asymptotic normality for intermediate order statistics, and derive explicit formulae for the normalising constants for tempered stable processes. We adapt Hill’s estimator of the tail index to the small time setting and establish its asymptotic normality under second order regular variation conditions, illustrating this with simulations. Applications to the fine structure of asset returns processes, possibly with infinite variation, are indicated.

Appendices

Appendix:

A Inverse of monotone functions

We employ the conventions (x,x) = ∅ for $-\infty \le x \le \infty$, and the infimum of the empty set is taken as $\infty$ (see Resnick (1987), Sect 0.2 for properties of inverse functions). Let τ_ℓ and τ_r be such that $-\infty \le \tau _{\ell }<\tau _{r}\le \infty$.

Define left (resp., right)-continuous inverses of monotone functions as follows: when $f:(l,r)\to \mathbb {R}$ is a non-decreasing (resp., non-increasing) function, set

$$ f^{\leftarrow,(l,r)}(y)=\inf\{x\in(l,r):f(x)\ge y \ \text{(resp.,}\ f(x) \le y)\},\ y\in\mathbb{R} . $$

(A.1)

For l < x_l < x_r < r and a monotone function $f:(l,r)\to \mathbb {R}$, note

$$ f^{\leftarrow,(l,r)}(y)=f^{\leftarrow,(x_{l},x_{r})}(y) $$

(A.2)

for all $y\in \mathbb {R}$, provided also f(x_l+) ∧ f(x_r−) < y < f(x_l+) ∨ f(x_r−).

Let ${\mathcal {C}}(f)$ be the continuity points of a function f. We need a result ensuring continuity of the inversion (see de Haan and Ferreira (2006) and Resnick (1987), Lemma 1.1.1 and Proposition 0.1, respectively).

Lemma A.1

Let $-\infty \le l<r\le \infty$. For $n=0,1,2,\dots$ let $f_{n}:(l,r)\to \mathbb {R}$ be non-decreasing or non-increasing. If $\lim _{n\to \infty }f_{n}(x)=f_{0}(x)$ for all $x\in (l,r)\cap {\mathcal {C}}(f_{0})$ then also $\lim _{n\to \infty }f_{n}^{\leftarrow }(y)=f_{0}^{\leftarrow }(y)$ for all $y\in J\cap {\mathcal {C}}(f_{0}^{\leftarrow })$, where

$$J := \big(f_{0}(l+)\wedge f_{0}(r-),f_{0}(l+)\vee f_{0}(r-)\big) \subseteq \mathbb{R} .$$

B Regular variation

A function $f:(0,r)\to \mathbb {R}$ is called regularly varying (Bingham et al. 1989; Geluk and de Haan 1987) with index $\alpha \in \mathbb {R}$ at 0, notation $f\in RV_{\alpha }^{0}$, provided f is a Borel function, satisfying

$$\lim_{t\downarrow 0}f(ut)/f(t)=u^{\alpha} ,\qquad u>0 .$$

A function g is called regularly varying with index α at $\infty$, denoted $g\in RV_{\alpha }^{\infty }$, provided $t\mapsto f(t):=g(1/t)\in RV_{-\alpha }^{0}$. If α = 0 the functions are said to be slowly varying.

C Cauchy functional equation

We shall employ the following variant of the Cauchy functional equation (for a proof see de Haan and Ferreira (2006), pp.7–8).

Lemma C.1

Let $f,h:(0,\infty )\to \mathbb {R}$ be functions such that 0∉range(h). Assume that f(1) = 0, f is monotone and f is non-constant.

If f(xy) = f(x)h(y) + f(y) for all x,y > 0 then there exists $\gamma \in \mathbb {R}$ such that f(x) = f^′(1)(x^γ − 1)/γ for x > 0.