PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 80(94) (2006), 109–120 DOI:10.2298/PIM0694109F

THIRD ORDER EXTENDED REGULAR VARIATION

Isabel Fraga Alves, Laurens de Haan, and Tao Lin

Abstract. A theory of third order extended regular variation is developed, somewhat similar to the corresponding theories of first and second order.

1. Introduction

The theory of regularly varying functions initiated by J. Karamata (1930, 1933) has turned out to be very useful and in fact indispensable in the probabilistic and statistical theory of extreme values.

An extended form of regular variation serves as necessary and sufficient condition for the domain of attraction of extreme value distributions. The same conditions are sufficient for the consistency of the estimators of the extreme value index γ.

Second order extended regular variation has proven very useful (almost indispensable) for establishing asymptotic normality of estimators of γ, in particular for calculating the asymptotic variance and bias. In second order extended regular variation an extra parameter shows up that we call ρ. Then asymptotic bias depends on this parameter.

One way of reducing the (asymptotic) bias of an estimator is subtracting the estimated bias from the estimator. In order to do so one needs an estimator for the second order parameter ρ. Such estimators have been developed. They are generally consistent under second order extended regular variation. But for proving asymptotic normality of a ρ-estimator third order extended regular variation comes into play.

It has been proved (Fraga Alves, de Haan, and Lin, 2003) that indeed third order extended regular variation is sufficient for the asymptotic normality of ρestimators. The cited paper also gives a sketch of the third order theory but this sketch had to be so short that it is extremely difficult to understand.

This paper offers full proofs of the main results for third order extended regular variation.

2000 Mathematics Subject Classification: Primary 26A12, 62G32.

Key words and phrases: Regular variation; third-order condition.

Partially supported by ERAS project FCT / POCI / FEDER.

2. The results

A measurable function f is said to satisfy the extended regular variation property if there is a positive function a such that for x > 0,

(2.1)  lim_{t→∞} \frac{f(tx) − f(t)}{a(t)} = \frac{x^{γ} − 1}{γ},

where γ is a real parameter (Notation f ∈ ERV or f ∈ ERV_{γ}).

The speed of convergence in this limit relation can be captured by a relation of second order. The measurable function f is said to satisfy the second extended regular variation property if there exist functions a > 0 and A, positive or negative, with lim_{t→∞} A(t) = 0, such that for all x > 0

\begin{aligned}lim_{t→∞} \frac{1}{A(t)} (\frac{f(tx) − f(t)}{a(t)} − \frac{x^{γ} − 1}{γ}) & = ∫_{1}^{x} y^{γ−1} ∫_{1}^{y} u^{ρ−1}du dy \\ & = \frac{1}{ρ} (\frac{x^{γ+ρ} − 1}{γ + ρ} − \frac{x^{γ} − 1}{γ}) , \end{aligned} ; (2.2)

where ρ ⩽ 0 is the second order parameter (Notation f ∈ 2ERV or f ∈ 2ERV_{γ,ρ}).

Now the third order relation becomes obvious. The function f satisfies the third order extended regular variation property if it satisfies (2.2) and there exists a positive or negative function B, with lim_{t→∞} B(t) = 0, such that for all x > 0

(2.3) lim_{t→∞} \frac{1}{B(t)} [\frac{1}{A(t)} (\frac{f(tx) − f(t)}{a(t)} − \frac{x^{γ} − 1}{γ}) − ∫_{1}^{x} y^{γ−1} ∫_{1}^{y} u^{ρ−1}du dy]

= ∫_{1}^{x} y^{γ−1} ∫_{1}^{y} u^{ρ−1} ∫_{1}^{u} s^{η−1}ds du dy,

where η ⩽ 0 is the third order parameter (Notation f ∈ 3ERV or f ∈ 3ERV_{γ,ρ,η}).

One of the results established in the Theorem is that only in trivial cases the limit function in (2.3) is not of the stated form.

Theorem 2.1. Write

D_{γ}(x) := ∫_{1}^{x} y^{γ−1}dy = \frac{x^{γ} − 1}{γ}  and
H_{γ,ρ}(x) := ∫_{1}^{x} y^{γ−1} ∫_{1}^{y} u^{ρ−1}du dy = \frac{1}{ρ} (\frac{x^{γ+ρ} − 1}{γ + ρ} − \frac{x^{γ} − 1}{γ}) .

Suppose f is a measurable function and there exist functions a_{0} (positive) and a_{1} and a_{2} (positive or negative) with lim_{t→∞} a_{i}(t) = 0 for i = 1, 2 such that for all x > 0

(2.4)  lim_{t→∞} {f(tx) − f(t) − a_{0}(t)D_{γ}(x) − a_{1}(t)H_{γ,ρ}(x)}/a_{2}(t) =: R(x),

exists (t → ∞), then for a judicious choice of a_{0}, a_{1} and a_{2} and given that D, H and R are not linearly dependent, the limit function R will be of the form

(2.5)  R_{γ,ρ,η}(x) := ∫_{1}^{x} y^{γ−1} ∫_{1}^{y} u^{ρ−1} ∫_{1}^{u} s^{η−1}ds du dy,

THIRD ORDER EXTENDED REGULAR VARIATION 111

with ρ ⩽ 0, η ⩽ 0. Moreover,

lim_{t→∞} \frac{a_{2}(tx)}{a_{2}(t)} = x^{γ+ρ+η}, (2.6)

lim_{t→∞} \frac{a_{1}(tx) − a_{1}(t)x^{γ+ρ}}{a_{2}(t)} = x^{γ+ρ} · \frac{x^{η} − 1}{η}, (2.7)

lim_{t→∞} \frac{a_{0}(tx) − a_{0}(t)x^{γ} − a_{1}(t)x^{γ} · (x^{ρ} − 1)/ρ}{a_{2}(t)} = x^{γ} · H_{ρ,η}(x) (2.8)

and for each ϵ > 0 there exists t_{0} such that for t ⩾ t_{0}, tx ⩾ t_{0} (2.9)

e^{−ϵ| log x|}x^{−γ−ρ−η} |\frac{f(tx) − f(t) − a_{0}(t)D_{γ}(x) − a_{1}(t)H_{γ,ρ}(x)}{a_{2}(t)} − R_{γ,ρ,η}(x)| ⩽ ϵ.

Proof. We write for t, x, y > 0

f(txy) − f(t) = (f(txy) − f(tx)) + (f(tx) − f(t))

and we connect the three parts to (2.4) as follows

(2.10) \begin{matrix}\{f(txy) − f(t) − a_{0}(t)D_{γ}(xy) − a_{1}(t)H_{γ,ρ}(xy)\}/a_{2}(t) \\ = [\{f(txy) − f(tx) − a_{0}(tx)D_{γ}(y) − a_{1}(tx)H_{γ,ρ}(y)\}/a_{2}(tx)]a_{2}(tx)/a_{2}(t) \\ + \{f(tx) − f(t) − a_{0}(t)D_{γ}(x) − a_{1}(t)H_{γ,ρ}(x)\} /a_{2}(t) \\ + \{−a_{0}(t) (D_{γ}(xy) − D_{γ}(x)) + a_{0}(tx)D_{γ}(y)\} /a_{2}(t) \\ + \{−a_{1}(t) (H_{γ,ρ}(xy) − H_{γ,ρ}(x)) + a_{1}(tx)H_{γ,ρ}(y)\} /a_{2}(t). \end{matrix}

Note that

−a_{0}(t) (D_{γ}(xy) − D_{γ}(x)) + a_{0}(tx)D_{γ}(y) = (a_{0}(tx) − x^{γ}a_{0}(t)) D_{γ}(y) and

− a_{1}(t) (H_{γ,ρ}(xy) − H_{γ,ρ}(x)) + a_{1}(tx)H_{γ,ρ}(y)

= (a_{1}(tx) − x^{γ+ρ}a_{1}(t)) H_{γ,ρ}(y) − a_{1}(t) x^{γ} \frac{x^{ρ} − 1}{ρ} \frac{y^{γ} − 1}{γ}

so that the sum of the last two terms of (2.10) is

(2.11) \frac{a_{1}(tx) − a_{1}(t)x^{γ+ρ}}{a_{2}(t)} H_{γ,ρ}(y) + \frac{a_{0}(tx) − a_{0}(t)x^{γ} + a_{1}(t)x^{γ}(x^{ρ} − 1)/ρ}{a_{2}(t)}D_{γ}(y).

Let t → ∞ in both sides of the equation (2.10) with the last two terms substituted by (2.11). We get for x, y > 0

R(xy) − R(x) = lim_{t→∞} [R(y)(1 + o(1))\frac{a_{2}(tx)}{a_{2}(t)} + H_{γ,ρ}(y)\frac{a_{1}(tx) − a_{1}(t) · x^{γ+ρ}}{a_{2}(t)}

(2.12)  + D_{γ}(y)\frac{a_{0}(tx) − a_{0}(t)x^{γ} − a_{1}(t)x^{γ} · (x^{ρ} − 1)/ρ}{a_{2}(t)}].

We have required that there do not exist constants c_{2} and c_{3} such that R(y) ≡ c_{2}H_{γ,ρ}(y) + c_{3}D_{γ}(y).

Consequently, the set of vectors {(R(y), H(y), D(y))}_{y>0} is not contained in a plane, hence there are y_{1}, y_{2}, y_{3} such that the matrix

(\begin{matrix}R(y_{1}) & H(y_{1}) & D(y_{1}) \\ R(y_{2}) & H(y_{2}) & D(y_{2}) \\ R(y_{3}) & H(y_{3}) & D(y_{3}) \end{matrix})

has rank 3. Then also the transposed matrix has rank 3. So there are no z_{1}, z_{2}, z_{3}, not all of them zero, such that ∑_{i} z_{i}R(y_{i}) = ∑_{i} z_{i}H(y_{i}) = ∑_{i} z_{i}D(y_{i}) = 0.

Now take z_{1}, z_{2}, z_{3} such that
(2.13)  ∑_{i} z_{i}H(y_{i}) = ∑_{i} z_{i}D(y_{i}) = 0;

then we must have ∑_{i} z_{i}R(y_{i}) ≠ 0.

Now multiply the two sides of (2.12) by z_{i}, i = 1, 2, 3, and add the three equations. The resulting equation (use (2.13)) shows that lim_{t→∞} a_{2}(tx)/a_{2}(t) exists for x > 0. The limit could be zero (but that is not possible here) or else a power of x, say x^{γ+ρ+η}. This is the definition of η. Similarly (repeating the above reasoning for this case)

(2.14)  lim_{t→∞} \frac{a_{1}(tx) − a_{1}(t) · x^{γ+ρ}}{a_{2}(t)}

and

(2.15)  lim_{t→∞} \frac{a_{0}(tx) − a_{0}(t) · x^{γ} − a_{1}(t) · x^{γ} · (x^{ρ} − 1)/ρ}{a_{2}(t)}

must exist. The limits in (2.14) and (2.15) must be [de Haan and Stadtmüller 1996, Theorem 1]

c_{1}x^{γ+ρ} · (x^{η} − 1)/η and x^{γ} (c_{1}H_{ρ,η}(x) + c_{2}D_{ρ}(x)) .

By replacing a_{2} with c_{1}a_{2} and a_{1} with a_{1} + c_{2}a_{2} in (2.4) the limits become x^{γ+ρ} · (x^{η} − 1)/η and x^{γ}H_{ρ,η}(x), respectively.

Now (2.12) leads to the functional equation (x, y > 0)

(2.16) x^{γ+ρ+η}R(y) = R(xy) − R(x) − x^{γ} · H_{ρ,η}(x)\frac{y^{γ} − 1}{γ} − x^{γ+ρ} · H_{γ,ρ}(y) · \frac{x^{η} − 1}{η}.

The reader may want to check that the function R_{γ,ρ,η} is a solution of (2.16). Let R′ be any other solution and define V := R_{γ,ρ,η} −R′. Then V satisfies the equation

x^{γ+ρ+η}V (y) = V (xy) − V (x),

hence (as in de Haan 1970, section 1.4),

V (x) = c · \frac{x^{γ+ρ+η} − 1}{γ + ρ + η}.

Next we get rid of V. Suppose the limit in (2.4) is R′ := R_{γ,ρ,η} + V. Note that by changing the functions a_{0}, a_{1} and a_{2} in (2.4) into

(2.17)  \{ \begin{matrix}ã_{2} := c_{2}^{−1}a_{2} \\ ã_{1} := a_{1} + c_{1}ã_{2} \\ ã_{0} := a_{0} + c_{0}ã_{2} \end{matrix} ,

the limit function R changes into c_{2}R−c_{0}D_{γ} −c_{1}H_{γ,ρ}. Also note that if both η ≠ 0 and η + ρ ≠ 0,

R_{γ,ρ,η}(x) = \frac{1}{η(η + ρ)} \frac{x^{γ+ρ+η} − 1}{γ + ρ + η} − \frac{1}{η} H_{γ,ρ}(x) − \frac{1}{η(η + ρ)}D_{γ}(x),

i.e.,

\frac{x^{γ+ρ+η} − 1}{γ + ρ + η} = η(η + ρ) \{R_{γ,ρ,η}(x) + \frac{1}{η} H_{γ,ρ}(x) + \frac{1}{η(η + ρ)}D_{γ}(x)\} .

If η = 0, then

\frac{x^{γ+ρ+η} − 1}{γ + ρ + η} = ρH_{γ,ρ}(x) + D_{γ}(x)

and if η + ρ = 0, then

\frac{x^{γ+ρ+η} − 1}{γ + ρ + η} = D_{γ}(x)

Hence by the device (2.17) we can make sure that the limit function in (2.4) is exactly R_{γ,ρ,η} .

In order to prove (2.9) we distinguish various cases.

1. γ = ρ = η = 0. The proof is similar to Omey and Willekens (1988). For x, y > 0

f(txy) − f(ty) − f(tx) − f(t) − a_{1}(t)(log x)(log y)
= f(txy) − f(t) − a_{0}(t) log(xy) − a_{1}(t)\frac{1}{2} (log(xy))^{2}
− \{f(ty) − f(t) − a_{0}(t) log y − a_{1}(t)\frac{1}{2} (log(y))^{2}\}
− \{f(tx) − f(t) − a_{0}(t) log x − a_{1}(t)\frac{1}{2} (log(x))^{2}\} .

Hence by (2.4) and (2.5) for the function g_{x}(t) := f(tx) − f(t), we have

lim_{t→∞} \frac{g_{x}(ty) − g_{x}(t) − a_{1}(t) log x log y}{a_{0}(t)} = \frac{1}{6} (log(xy))^{3} − \frac{1}{6} (log y)^{3} − \frac{1}{6} (log x)^{3}
= \frac{1}{2} (log x)^{2} log y + \frac{1}{2} (log y)^{2} log x

for all x, y > 0, i.e.,

lim_{t→∞} \frac{g_{x}(ty) − g_{x}(t) − \{a_{1}(t)(log x) − a_{0}(t)\frac{1}{2} (log x)^{2}\} log y}{a_{0}(t) log x} = \frac{1}{2} (log y)^{2} .

Hence g_{x} is second order Π-varying for each x > 0. It follows from Omey and Willekens (1988), that the function h_{x}(t) := g_{x}(t) − \frac{1}{t} ∫_{0}^{t} g_{x}(s) ds is in the class Π with auxiliary function a_{0}(t) log x, i.e., for y > 0

lim_{t→∞} \frac{h_{x}(ty) − h_{x}(t)}{a_{0}(t) log x} = log y.

The theory of Π-variation applies, hence by Geluk and de Haan (1987)

(2.18)  lim_{t→∞} \frac{1}{a_{0}(t) log x}(h_{x}(t) − \frac{1}{t} ∫_{0}^{t} h_{x}(s)ds) = 1.

Now note that

h_{x}(t) = h^{(1)}(tx) − h^{(1)}(t) with h^{(1)}(t) := f(t) − \frac{1}{t} ∫_{0}^{t} f(u) du.

Consequently, it follows from (2.18) that

lim_{t→∞} \frac{1}{a_{0}(t) log x}(h^{(1)}(tx) − h^{(1)}(t) − \frac{1}{tx} ∫_{0}^{tx} h^{(1)}(s) ds + \frac{1}{t} ∫_{0}^{t} h^{(1)}(s) ds) = 1,

i.e.,

(2.19)  lim_{t→∞} \frac{h^{(2)}(tx) − h^{(2)}(t)}{a_{0}(t) log x} = 1

with

h^{(2)}(t) := h^{(1)}(t) − \frac{1}{t} ∫_{0}^{t} h^{(1)}(s)ds.

Again by the theory of Π-variation, relation (2.19) implies

lim_{t→∞} \frac{h^{(2)}(tx) − h^{(2)}(t)}{h^{(3)}(t)} = log x

with

h^{(3)}(t) := h^{(2)}(t) − \frac{1}{t} ∫_{0}^{t} h^{(2)}(s)ds.

Then by Drees’ inequalities (1998), for any ϵ_{1}, ϵ_{2} > 0, there exists t_{0} = t_{0}(ϵ_{1}, ϵ_{2}) such that for t ⩾ t_{0}, tx ⩾ t_{0}

(2.20)  e^{−ϵ_{2}| log x|} |\frac{h^{(2)}(tx) − h^{(2)}(t)}{h^{(3)}(t)} − log x| < ϵ_{1}.

It is easily checked using the definition of h^{(2)} that

h^{(1)}(t) = h^{(2)}(t) + ∫_{0}^{t} h^{(2)}(s)\frac{ds}{s}

hence

h^{(1)}(tx) − h^{(1)}(t) = h^{(2)}(tx) − h^{(2)}(t) + ∫_{1}^{x} h^{(2)}(ts)\frac{ds}{s},

i.e., (2.21)

h^{(1)}(tx) − h^{(1)}(t) − h^{(2)}(t) log x = h^{(2)}(tx) − h^{(2)}(t) + ∫_{1}^{x} (h^{(2)}(ts) − h^{(2)}(t)) \frac{ds}{s}

and consequently

\frac{h^{(1)}(tx) − h^{(1)}(t) − (h^{(2)}(t) + h^{(3)}(t)) log x}{h^{(3)}(t)} − \frac{(log x)^{2}}{2}

= \frac{h^{(2)}(tx) − h^{(2)}(t)}{h^{(3)}(t)} − log x + ∫_{1}^{x} (\frac{h^{(2)}(ts) − h^{(2)}(t)}{h^{(3)}(t)} − log s) \frac{ds}{s}.

Integrating (2.21) with respect to x we get

(2.22) ∫_{1}^{x} (h^{(1)}(ts) − h^{(1)}(t)) \frac{ds}{s} − h^{(2)}(t) ∫_{1}^{x}(log s)\frac{ds}{s}

= ∫_{1}^{x} (h^{(2)}(ts) − h^{(2)}(t)) \frac{ds}{s} + ∫_{1}^{x} ∫_{1}^{u} (h^{(2)}(ts) − h^{(2)}(t)) \frac{ds}{s} \frac{du}{u}.

Also we have the following obvious analogue of (2.21): (2.23)

f(tx) − f(t) − h^{(1)}(t) log x = (h^{(1)}(tx) − h^{(1)}(t)) + ∫_{1}^{x} (h^{(1)}(ts) − h^{(1)}(t)) \frac{ds}{s}.

Combining (2.22) and (2.23) we get

f(tx) − f(t) = h^{(1)}(t) log x + h^{(2)}(t)(log x)^{2}/2

+ ∫_{1}^{x} ∫_{1}^{u} (h^{(2)}(ts) − h^{(2)}(t)) \frac{ds}{s} \frac{du}{u} + ∫_{1}^{x} (h^{(2)}(ts) − h^{(2)}(t)) \frac{ds}{s}

+ (h^{(1)}(tx) − h^{(1)}(t)) ,

which by (2.21) equals

(h^{(1)}(t) + h^{(2)}(t)) log x + h^{(2)}(t)(log x)^{2}/2 + ∫_{1}^{x} ∫_{1}^{u} (h^{(2)}(ts) − h^{(2)}(t)) \frac{ds}{s} \frac{du}{u}

+ 2 ∫_{1}^{x} (h^{(2)}(ts) − h^{(2)}(t)) \frac{ds}{s} + h^{(2)}(tx) − h^{(2)}(t).

Hence

\frac{f(tx) − f(t) − (h^{(1)}(t) + h^{(2)}(t) + h^{(3)}(t)) log x − (h^{(2)}(t) + 2h^{(3)}(t)) (log x)^{2}/2}{h^{(3)}(t)}

− \frac{(log x)^{3}}{6} = ∫_{1}^{x} ∫_{1}^{u} (\frac{h^{(2)}(ts) − h^{(2)}(t)}{h^{(3)}(t)} − log s) \frac{ds}{s} \frac{du}{u}

+ 2 ∫_{1}^{x} (\frac{h^{(2)}(ts) − h^{(2)}(t)}{h^{(3)}(t)} − log s) \frac{ds}{s} + \frac{h^{(2)}(tx) − h^{(2)}(t)}{h^{(3)}(t)} − log x.

Hence by (2.20) for any ϵ_{1}, ϵ_{2} > 0, t ⩾ t_{0}(ϵ_{1}, ϵ_{2}), tx ⩾ t_{0}(ϵ_{1}, ϵ_{2})

|\frac{f(tx) − f(t) − (h^{(1)}(t) + h^{(2)}(t) + h^{(3)}(t)) log x − (h^{(2)}(t) + 2h^{(3)}(t)) (log x)^{2}/2}{h^{(3)}(t)}
−\frac{(log x)^{3}}{6} |

< ϵ_{1} ∫_{1}^{x} ∫_{1}^{u} e^{ϵ_{2}|log s|} \frac{ds}{s} \frac{du}{u} + 2ϵ_{1} ∫_{1}^{x} e^{ϵ_{2}|log s|} \frac{ds}{s} + ϵ_{1}e^{ϵ_{2}|log x|}

< ϵ_{1} (1 + \frac{1}{ϵ_{2}} + \frac{1}{ϵ_{2}^{2}} ) e^{ϵ_{2}|log x|}.

2. η < 0: By Theorem 1.10 of Geluk and de Haan (1987) relation (2.7), that is,

lim_{t→∞} \frac{(tx)^{−(γ+ρ)}a_{1}(tx) − t^{−(γ+ρ)}a_{1}(t)}{t^{−(γ+ρ)}a_{2}(t)} = \frac{x^{η} − 1}{η}

with η < 0, implies that for some constant c > 0

c − t^{−(γ+ρ)}a_{1}(t) ∼ −η^{−1}t^{−(γ+ρ)}a_{2}(t),

that is, a_{1}(t) = ct^{γ+ρ} + η^{−1}a_{2}(t)(1 + o(1)) as t → ∞. Hence we can replace a_{1} in (2.8) with a_{1}^{∗} defined by a_{1}^{∗}(t) := ct^{γ+ρ} + η^{−1}a_{2}(t).

By inserting this choice in (2.8) and rearranging we get for x > 0

lim_{t→∞} \frac{a_{0}(tx) − a_{0}(t)x^{γ} − ct^{γ+ρ}x^{γ} · (x^{ρ} − 1)/ρ}{a_{2}(t)} = x^{γ} · H_{ρ,η}(x) − η^{−1}x^{γ} · \frac{x^{ρ} − 1}{ρ}

which is the same as

lim_{t→∞} \frac{(tx)^{−γ}a_{0}(tx) − t^{−γ}a_{0}(t) − ct^{ρ} · (x^{ρ} − 1)/ρ}{t^{−γ}a_{2}(t)} = \frac{1}{η} · \frac{x^{ρ+η} − 1}{ρ + η}.

We can rewrite this as

lim_{t→∞} \frac{((tx)^{−γ}a_{0}(tx) − c((tx)^{ρ} − 1)/ρ) − (t^{−γ}a_{0}(t) − c(t^{ρ} − 1)/ρ)}{t^{−γ}a_{2}(t)} = \frac{1}{η} · \frac{x^{ρ+η} − 1}{ρ + η},

where ρ + η < 0. Again by Theorem 1.10 of Geluk and de Haan (1987) this implies that for some constant c_{1},

c_{1} − t^{−γ}a_{0}(t) − c\frac{t^{ρ}}{ρ} ∼ \frac{1}{η(ρ + η)}t^{−γ}a_{2}(t), t → ∞,

hence in (2.4) we can replace a_{0} with a_{0}^{∗} defined by

a_{0}^{∗}(t) := c_{1}t^{γ} + \frac{c}{ρ}t^{γ+ρ} − \frac{1}{η(ρ + η)}a_{2}(t).

Inserting the special choices a_{1}^{∗} and a_{0}^{∗} in (2.4) we get

(2.24)  lim_{t→∞} \frac{1}{a_{2}(t)}[f(tx) − c_{1} \frac{(tx)^{γ} − 1}{γ} − \frac{c}{η} \frac{(tx)^{γ+ρ} − 1}{γ + ρ}

− (f(t) − c_{1} \frac{t^{γ} − 1}{γ} − \frac{c}{η} \frac{t^{γ+ρ} − 1}{γ + ρ}) − \frac{c}{η} \frac{t^{γ+ρ}}{η} \frac{x^{γ} − 1}{γ}]

= R_{γ,ρ,η}(x) + \frac{1}{η} H_{γ,ρ}(x) − \frac{1}{η(ρ + η)} \frac{x^{γ} − 1}{γ}

= \frac{1}{η} (H_{γ,ρ+η}(x) − \frac{1}{ρ + η} \frac{x^{γ} − 1}{γ}) = \frac{1}{η} \frac{x^{γ+ρ+η} − 1}{γ + ρ + η}.

Relation (2.24) is a second order relation for which uniform bounds are well known (Drees, 1998). Hence for some functions â_{1} and â_{2}

e^{−ϵ| log x|}x^{−γ−ρ−η}|[ (f(tx) − c_{1} \frac{(tx)^{γ} − 1}{γ} − \frac{c}{η} \frac{(tx)^{γ+ρ} − 1}{γ + ρ})

− (f(t) − c_{1} \frac{t^{γ} − 1}{γ} − \frac{c}{η} \frac{t^{γ+ρ} − 1}{γ + ρ}) − â_{1}(t)\frac{x^{γ} − 1}{γ}]\frac{1}{â_{2}(t)}| ⩽ ϵ

provided t ⩾ t_{0}(ϵ) and tx ⩾ t_{0}(ϵ). This gives (2.9) for η < 0.

3. ρ < 0, η = 0. We start from (2.8), i.e.,

lim_{t→∞} \frac{(tx)^{−γ}a_{0}(tx) − t^{−γ}a_{0}(t) − t^{−γ}a_{1}(t) · (x^{ρ} − 1)/ρ}{t^{−γ}a_{2}(t)} = H_{ρ,0}(x).

This relation is discussed in de Haan and Stadtmüller (1996), Theorem 2,(ii). Relation (2.11) of that paper gives

lim_{t→∞} \frac{\{(tx)^{−γ}a_{0}(tx) − \frac{1}{ρ}(tx)^{−γ}a_{1}(tx)\} − \{t^{−γ}a_{0}(t) − \frac{1}{ρ}t^{−γ}a_{1}(t)\}}{t^{−γ}a_{2}(t)} = −\frac{1}{ρ} · \frac{x^{ρ} − 1}{ρ}.

The latter relation is discussed in Proposition 2(ii) of the same paper. It follows that for some c > 0

lim_{t→∞} \frac{t^{−γ}a_{0}(t) − \frac{1}{ρ}t^{−γ}a_{1}(t) − c}{t^{−γ}a_{2}(t)} = −\frac{1}{ρ^{2}} ,

i.e., relation (2.4) holds with a_{1} replaced by a_{1}^{∗} defined by

a_{1}^{∗}(t) := −cρt^{γ} + ρa_{0}(t) + \frac{1}{ρ}a_{2}(t).

After some rearranging we get

lim_{t→∞} \frac{1}{a_{2}(t)} [(f(tx) − c\frac{(tx)^{γ} − 1}{γ}) − (f(t) − c\frac{t^{γ} − 1}{γ}) − (ct^{γ} − a_{0}(t)) \frac{x^{γ+ρ} − 1}{γ + ρ}]

= R_{γ,ρ,0}(x) + \frac{1}{ρ}H_{γ,ρ}(x) = \frac{1}{ρ}H_{γ+ρ,0}(x).

This is a second order relation for which uniform bounds are well known (Drees, 1998). This gives (2.9) for η = 0, ρ < 0.

118 FRAGA ALVES, DE HAAN, AND LIN

4. In the case γ ≠ 0, ρ = η = 0 we have, as t → ∞:

\frac{f(tx)−f(t)−a_{0}(t)(x^{γ} −1)/γ−a_{1}(t)H_{γ,0}(x)}{a_{2}(t)} → R_{γ,0,0}(x) = \frac{x^{γ} log^{2} x}{2γ} − \frac{1}{γ} H_{γ,0}(x).

Hence

\frac{f(tx) − f(t) − a_{0}(t)(x^{γ} − 1)/γ − (a_{1}(t) − a_{2}(t)/γ)H_{γ,0}(x)}{a_{2}(t)} → \frac{x^{γ} log^{2} x}{2γ}.

Note that

H_{γ,0}(x) = \frac{x^{γ} log x}{γ} − \frac{x^{γ} − 1}{γ^{2}}.

We have

\frac{1}{a_{2}(t)}[f(tx)−f(t)−(a_{0}(t)− \frac{1}{γ} a_{1}(t)+ \frac{1}{γ^{2}} a_{2}(t))\frac{x^{γ} − 1}{γ} −(a_{1}(t)− \frac{1}{γ} a_{2}(t))\frac{x^{γ} log x}{γ}]

→ \frac{x^{γ} log^{2} x}{2γ}.

Define

â_{0}(t) := a_{0}(t) − \frac{1}{γ} a_{1}(t) + \frac{1}{γ^{2}} a_{2}(t), â_{1}(t) := a_{1}(t) − \frac{1}{γ} a_{2}(t).

Then

\frac{1}{a_{2}(t)}[f(tx) − f(t) − â_{0}(t)\frac{x^{γ} − 1}{γ} − â_{1}(t)\frac{x^{γ} log x}{γ}] → \frac{x^{γ} log^{2} x}{2γ},

which can be rewritten as

\frac{(f(tx) − γ^{−1}â_{0}(tx)) − (f(t) − γ^{−1}â_{0}(t))}{a_{2}(t)} + \frac{â_{0}(tx) − â_{0}(t)x^{γ} − â_{1}(t)x^{γ} log x}{γa_{2}(t)}

→ \frac{x^{γ} log^{2} x}{2γ}. (2.25)

Relations (2.6), (2.7) and (2.8) imply

(2.26)  \frac{â_{0}(tx) − â_{0}(t)x^{γ} − â_{1}(t)x^{γ} log x}{a_{2}(t)} → \frac{x^{γ} log^{2} x}{2}.

From (2.25) and (2.26) we have

(2.27)  \frac{(f(tx) − γ^{−1}â_{0}(tx)) − (f(t) − γ^{−1}â_{0}(t))}{a_{2}(t)} → 0.

• If γ < 0, by Theorem 3.1.10 of Bingham, Goldie and Teugels (1987, due to Ash, Erdös and Rubel, 1974), the limit

c := lim_{t→∞} f(t) − γ^{−1}â_{0}(t)

exists, finite, and

lim_{t→∞} \frac{c − f(t) + γ^{−1}â_{0}(t)}{a_{2}(t)} = 0,

i.e.,

â_{0}(t) = γ (f(t) − c) + o(a_{2}(t)),

as t → ∞. Combining this with (2.26) gives

(2.28)  \frac{γf(tx) − γf(t)x^{γ} − â_{1}(t)x^{γ} log x}{a_{2}(t)} → \frac{x^{γ} log^{2} x}{2},

i.e.,

\frac{(tx)^{−γ}f(tx) − t^{−γ}f(t) − â_{1}(t)x^{γ}(log x)/γ}{a_{2}(t)/γ} → \frac{log^{2} x}{2}.

Note that this is just a second order condition for which uniform inequalities have been derived in Drees (1998). They can be rewritten as follows: one can find functions â_{1}^{∗} and â_{2}^{∗} such that for any ϵ > 0, t, tx ⩾ t_{0}(ϵ),

(2.29) x^{−γ} exp{−| log x|ϵ}

× |\frac{1}{â_{2}^{∗}(t)}[γf(tx) − γf(t) − γ^{2}f(t)\frac{x^{γ} − 1}{γ} − â_{1}^{∗}(t)x^{γ} log x] − \frac{x^{γ} log^{2} x}{2}| < ϵ.

• For γ > 0, Theorem 3.1.12 of of Bingham, Goldie and Teugels (1987, due to Bojanic and Karamata, 1963), applied to (2.27) implies

\frac{f(t) − γ^{−1}â_{0}(t)}{a_{2}(t)} → 0,

i.e.,

â_{0}(t) = γf(t) + o(a_{2}(t)),

as t → ∞. Combining this with (2.26) gives (2.28) once again and hence (2.29) is obtained. □

References

[1] J. M. Ash, P. Erdös, P. and L. A. Rubel, Very slowly varying functions, Aeq. Math. 10 (1974), 1–9.

[2] N. Bingham, C. Goldie, and J. Teugels, Regular Variation, Cambridge University Press, New York, 1987.

[3] R. Bojanić and J. Karamata, On a Class of Functions of Regular Asymptotic Behaviour, Math. Research Center Tech. Report 436 (1963), Madison, Wis.

[4] A. L. M. Dekkers, J. H. J. Einmahl and L. de Haan, A moment estimator for the index of an extreme-value distribution, Ann. Statist. 17 (1989), 1833–1855.

[5] H. Drees, On smooth statistical tail functionals, Scandinavian J. Stat. 25 (1998), 187–210.

[6] M. I. Fraga Alves, L. de Haan and T. Lin, Estimation of the parameter controlling the speed of convergence in extreme value theory, Math. Methods of Statist. 12 (2003), 155–176.

[7] J. Geluk and L. de Haan, Regular Variation, Extensions and Tauberian Theorems, CWI Tract 40, Center for Mathematics and Computer Science, Amsterdam, Netherlands, 1987.

[8] L. de Haan, On regular variation and its application to the weak convergence of sample extremes, Thesis, University of Amsterdam / Mathematical Centre tract 32, 1970.

[9] L. de Haan and U. Stadtmüller, Generalized regular variation of second order, J. Austral. Math. Soc. (A) 61 (1996), 381–395.

[10] J. Karamata, Sur un mode de croissance régulière des fonctions, Mathematica (Cluj) 4 (1930), 38–53.

[11] J. Karamata, Sur un mode de croissance régulière. Théorèmes fondamentaux, Bull. Soc. Math. France 61 (1933), 55–62.

[12] E. Omey and E. Willekens, π-variation with remainder, J. London Math. Soc. 37 (1988), 105–118.

CEAUL and DEIO Faculty of Science (Received 13 03 2006) University of Lisbon Portugal

isabel.alves@fc.ul.pt

Department of Economics Faculty of Economics Erasmus University Rotterdam The Netherlands

ldhaan@few.eur.nl

Xiamen University Department of Mathematics Xiamen University China

weiminghult@tom.com