1. INTRODUCTION
The Gaussian (Normal) curve is well known as the universal probability
law governing the statistical distribution of large samples. However, what
truly impacts us is not the overwhelming majority of “normal
events,” but the few rare exceptions of “abnormal”
extreme events. Indeed, it is the unique Michael Jordan we remember,
rather than a multitude of excellent professional NBA basketball players
scoring an average number of points per game; it is hurricane Andrew that
insurance companies (painfully) recall, rather than the thousands of
strong storms that took place in the United States in the 1990s; and it is
the 1912 Titanic disaster we remember, rather than numerous other
unfortunate sinking events of ships in the Atlantic. For many more
examples of the importance we attribute to extreme events, we refer the
readers to the Guinness Book of Records.
The interest in rare and extreme events was shared—apart from
the devoted readers of the Guinness Book of Records—also by
the scientific community. The pioneering studies took place in the 1920s
and 1930s, with the works of von Bortkiewicz [20], Fréchet [8], Fisher and Tippett [7], von Mises [21], Weibull, and Gumbel [12]. A rigorous theoretical framework was
presented in 1943 by Gnedenko [11].
Nowadays, the study of extremes is a well-established branch of
Probability Theory called Extreme Value Theory (EVT). This theory
is of major importance in the analysis of rare and
“catastrophic” events such as floods in hydrology, large
claims in insurance, crashes in finance, material failure in corrosion
analysis, and so forth. Classic references on EVT are [9,12]. For both the
theory and applications of modern EVT we refer the readers to [3,15,19].
Given a sequence
{ξn}n=1∞ of
independent and identically distributed (i.i.d.) random variables (random
samples), the “normal” approach is to study the asymptotic
behavior of the scaled sums of the ξ's, namely the limiting
probability distribution of
(as n → ∞), where
{an}n=1∞
and
{bn}n=1∞
are properly chosen scaling coefficients. This, as the Central Limit
Theorem asserts, leads to the universal Gaussian (Normal) distribution.
(In case the ξ's fail to have a finite variance or mean, the
Central Limit Theorem is replaced by the Generalized Central Limit
Theorem, and the limiting distributions are the stable Lévy laws
[13].) The “extreme”
approach, on the other hand, studies the asymptotic behavior of the scaled
maxima of the ξ's:
The “Central Limit Theorem” of EVT asserts that (2) has
three possible limiting probability laws (named, respectively, after the
pioneers of EVT): Fréchet, Weibull, and Gumbel. The probability
distribution functions of these universal extreme value laws are
(the exponent α appearing in the Fréchet and Weibull
distributions is a positive parameter).
Typically, EVT studies the extreme statistics (maxima, minima, order
statistics, records, etc.) of random sequences
{ξn}n=1∞. The
fundamental theory considers i.i.d. sequences, but generalizations to
stationary sequences do exist (see, for example, [3]). Such random sequences represent a time
series of data measured at discrete-time epochs. However,
most “real-world” systems are usually continuous-time
systems. Hence, why not study the extremes of continuous-time systems
directly in some appropriate continuous-time setting?
Consider a generic continuous-time physical system in which events
that take place are monitored and logged. Each event is described by a
pair (t,x) of coordinates, with t being the
time at which the event occurred and x being the
magnitude of the event (a numerical value). Hence, the
“history” of the physical system is given by a random
collection
of points in the plane, where each point of
represents an event that took place. In the mathematical nomenclature,
the random collection
—the system's history of events—is called a
point process. A direct continuous-time approach would thus be to
study the statistics of the extreme points of
.
To that end, one obviously has to specify the probability distribution of
the point process
.
The continuous-time counterparts of discrete-time i.i.d. sequences are
Poisson point processes. If
is a Poisson point process governed by the rate function
λ(x), then, informally, (i) events of magnitude belonging to the
infinitesimal range (x,x + dx) arrive at rate
λ(x)dx and (ii) the occurrences of events of
different magnitudes are independent.
In this article, we will explore the extremes of a generic
continuous-time physical system whose history of events forms a Poisson
point process. As we will demonstrate, this setting turns out to be
“tailor-made” to the modeling and analysis of extreme events
in continuous time. We begin, in Section 2, with a short review of the
notion of Poisson point processes and introduce our underlying
continuous-time system model. In Section 3, we define the system's
sequence of order statistics and explore their distributions: the
probability law of the maximum, the probability law of the nth
“runner-up,” and the multidimensional probability law of the
“top n” extremes. In Section 4, we unveil a hidden
Poissonian structure underlying the sequence of order statistics.
This hidden structure gives rise to a markedly simple simulation
algorithm for the sequence of order statistics. In Section 5, we turn
to study the internal hierarchy of the sequence of order
statistics; namely we analyze the magnitudes of the extreme events when
measured relative to each other. We conclude, in Section 6, with
the exploration of the system's records times and record
values.
A note about notation: Throughout the article,
denotes the real line, P(·) := probability,
E[·] := expectation, and dω
(dx, dt, etc.) is used to denote the infinitesimal
neighborhood of the point ω (x, t, etc.).
2. THE CONTINUOUS-TIME SETTING
In this section we concisely review the notion of Poisson point
processes, introduce our underlying continuous-time “event
process” (which will accompany us throughout the manuscript),
and explain the passage from the discrete-time i.i.d. setting to the
continuous-time Poisson setting.
2.1. Poisson Point Processes
Let Ω be an Euclidean space (or subspace or domain). A random
countable collection of points Π ⊂ Ω is called a point
process [17]. We denote by
Π(B) the number of points of Π residing in the subset
B:
Hence, the point process Π ⊂ Ω induces a counting
measure on Ω given by (4).
A point process Π ⊂ Ω is said to be a Poisson point
process [17] with rate
r(ω) if the following pair of conditions hold:
- If B is a subset of Ω, then the random variable
Π(B) is Poisson distributed with mean
∫B r(ω) dω.
- If {Bk}k is a
finite collection of disjoint subsets of Ω, then
{Π(Bk)}k is a finite
collection of independent random variables.
The Poisson point process Π can also be described by its
finite-dimensional distributions: The probability that a given set of
points
{ωj}j=1n
belongs to Π is
Informally, Ω is divided into infinitesimal cells. Each cell
contains either a single point or no points at all. The cells are
independent, and the probability that the cell dω contains a
point is r(ω) dω.
The best known example of a Poisson point process is the standard
Poisson process, where Ω = [0,∞) and the rate
function r(ω) is constant.
2.2. The Event Process
Equipped with the notion of Poisson point processes, we can now
rigorously define the point process
presented in Section 1. Recall that we considered a generic
continuous-time physical system whose “history” is a random
collection
of points in the plane—the point (t,x) represents
an event of magnitude x taking place at time t. In
Section 1, we said, informally, that “events of magnitude belonging
to the infinitesimal range (x,x + dx) arrive at
rate λ(x) dx” and, “the occurrences of
events of different magnitudes are independent.” Put rigorously, the
process
(henceforth referred to as our event process) is taken to be a
time-homogeneous Poisson point process on
with rate
Thus,
We set Λ(x) to be the Poissonian rate at which samples of
size greater than x arrive, namely
The function Λ(x) is smooth, monotone nonincreasing, and
fully characterizes the event process
.
This function will turn out to play a key role in the sequel. We
henceforth refer to the function Λ(x) as the
characteristic of the process
and assume that
We will elaborate on this assumption in the following subsection.
Furthermore, in Subsection 3.1 we shall show that this assumption is in
fact an essential requirement.
Finally, note that the random variable
—counting the number of events of size x ∈
[a,b] occurring during the time interval
[t,t + Δ] —is Poisson distributed
with mean
2.3. From Discrete to Continuous and Back
In the discrete-time setting described in Section 1, the underlying
time series
{ξn}n=1∞ is
an i.i.d. sequence of random variables. If we take the random samples
{ξn}n=1∞ to
arrive according to some continuous-time counting process
(N(t))t≥0, then the sample set,
at time t, would be
(the sample set being empty in the case N(t) = 0).
The setting in which (N(t))t≥0 is
a renewal process was introduced in [10] and coined “Random Record Models”
(see also [4,22]).
If (N(t))t≥0 is a standard
Poisson process with rate ρ, then the “sample process”
(11) is in fact a time-homogeneous Poisson point process on
with rate
where f (x) is the probability density function of
the ξ's.
On the other hand, if the rate function λ(x) of the event
process
is integrable (i.e., if ∫λ(x) dx < ∞),
then
can be described in the form of the sample set (11), where (i) the
Poissonian arrival rate is ρ := ∫λ(x) dx and
(ii) the probability density function of the ξ's is f
(x) := λ(x)/ρ.
Hence, in the case of integrable rate functions, there is a one-to-one
correspondence, via embedding, between discrete-time i.i.d. sequences
{ξn}n=1∞ and
continuous-time event processes
.
However, the truly interesting case is where the rate function is
actually not integrable: ∫λ(x) dx =
∞. In this case,
has infinitely many events occurring on all time scales
(i.e., on any time interval), and no correspondence between
and any discrete-time i.i.d. sequence
{ξn}n=1∞ can
be established. Thus, in the case of a nonintegrable rate function, the
event process
is a truly continuous-time “creature.”
In this article, we focus on the “truly continuous-time
case,” where the rate functions are nonintegrable. This is
the reason for assumption (9). The limit
limx→−∞ Λ(x) = +∞
ensures that the rate function λ(x) is indeed nonintegrable.
The limit limx→+∞ Λ(x) = 0, on
the other hand, ensures that events of infinitely large magnitude do not
occur. We will return to discuss assumption (9) in Subsection 3.1.
3. THE SEQUENCE OF ORDER STATISTICS
Let
denote the sequence of order statistics of the event process
;
namely Xn(t) is the nth
largest sample observed during the time period
[0,t]:
where X0(t) is set to equal +∞.
(Definition (13) is in fact valid for any point process on
.)
Let us now turn to analyze the probability distributions of the
maximum value X1(t), the nth
“runner-up” Xn+1(t), and
the vector of the “top n” order statistics
(X1(t),…,Xn(t)).
3.1. The Maximum
We compute the distribution of the maximum
X1(t). The maximum at time t is less
or equal to x if and only if no events of magnitude greater than
x occurred during the time period [0,t]; that
is,
However, the random variable
is Poisson distributed with mean tΛ(x) (recall
(10)) and, hence,
The probability density function of X1(t)
is given, in turn, by
The assumption of (9): From (14), we see that the distribution of the
maximum is proper if and only if the assumption of (9) holds. Indeed, in
order for the distribution of X1(t) to be
proper, the limit of (14) must equal zero when x →
−∞ and must equal one when x → +∞. This,
however, takes place if and only if
limx→−∞ Λ(x) = +∞
and limx→+∞ Λ(x) = 0. Hence,
the assumption of (9) is not an impeding restriction, but an essential
requirement.
3.2. Beating the Maximum
Assume that the current maximum level is x. How long will we
have to wait until this maximum level is beat? Since events of magnitude
greater than x arrive at rate Λ(x), the answer is
straightforward: The waiting time is exponentially distributed with rate
Λ(x) (mean 1/Λ(x)). Let us consider the two
following variations of this question.
(i) Assume that we are at time t, but we do
not know the current maximum level
X1(t). How long will we have to
wait—from the present time t onward—until the unknown
maximum level X1(t) is beat? Let
L(t) denote the respective waiting time. Given that
X1(t) = x, the waiting time
L(t) is exponentially distributed with rate
Λ(x). Conditioning on the (unknown) maximum level
X1(t), we obtain that
The proof of (16) is given below. Note that the waiting time
L(t) has infinite mean.
(ii) This variation regards positive-valued event processes.
Assume (as earlier) that the current maximum level is x and let
k > 1 be a fixed parameter. How long will we have to wait
until the occurrence of a maximum level that is at least k times
larger than all the maximum levels preceding it? These waiting
times (coined “geometric record times”) are explored in
[2] (their analysis is considerably
harder than the analysis of the waiting times described
earlier).
To prove (16), use the probability density function of (15) and the
change of variables u = tΛ(x):
3.3. Fréchet, Weibull, and Gumbel
Equation (14) implies that the function Λ(x) fully
characterizes the distribution of the maximum. On the other hand, the
function Λ(x) also fully characterizes the underlying event
process
.
Hence, there is a one-to-one correspondence—conveyed by
the characteristic Λ(x)—between event processes and the
distribution of their maxima. In particular, the event processes
corresponding to the “Central Limit Theorem” distributions of
EVT (viz. the Fréchet, Weibull, and Gumbel distributions) are
given, respectively, by
(the exponent α appearing in the Fréchet and Weibull
distributions is a positive parameter).
Could these extreme value laws be obtained (as in the
“classic” EVT) as the only possible maxima scaling limits? The
answer, as explained below, is affirmative.
The continuous-time scaling of the maximum
X1(t), analogous to the discrete-time scaling
of (2), is
where a(t) and b(t) are the
scaling functions (a(t) being positive valued). The
distribution of the scaled maxima
,
using (14), is hence given by
On the other hand, the distribution of the scaled maxima
of (2) is given by
where L(x) := −ln(P(ξ1
≤ x)). Moreover, the function L(x) satisfies
the very same properties the function Λ(x) does; namely it is
monotone decreasing, with limx→−∞
L(x) = +∞ and
limx→+∞ L(x) = 0.
Thus, (17) (as t → ∞) and (18) (as n
→ ∞) must yield the same distributional limits. The
“classic” EVT asserts that the three possible limits of (18)
are the extreme value distributions: Fréchet, Weibull, and Gumbel.
Hence, these probability laws are also the only possible limits of
(17).
The scaling functions in the Fréchet, Weibull, and Gumbel cases
are
It is interesting to note that in these three special cases, the
scaling yields
.
For a detailed analysis regarding the “basins of
attraction” of the Fréchet, Weibull, and Gumbel extreme value
distributions, we refer the readers to [3].
3.4. The nth “Runner-up” and the
“Top n”
The distribution of
Xn+1(t)—the nth
“runner-up” is computed analogously to the distribution of the
maximum X1(t). The (n + 1)st order
statistic at time t is less or equal to x if and only if
no more than n events of magnitude greater than x
occurred during the time period [0,t]; that is,
Hence, since
is Poisson distributed with mean tΛ(x), we have
The probability density function of
Xn+1(t) is given, in turn, by
Note that (19) and (20) indeed coincide with (14) and (15) in the case
n = 0.
Equation (19) gives the one-dimensional distribution of the
order statistics. What about the joint, multidimensional,
probability distribution of the order statistics; namely the joint
distribution of the “top n”—the vector
(X1(t),…,Xn(t))?
Well, the joint probability density function of the vector
(X1(t),···,Xn(t))
is given by
where x1 > x2 >
··· > xn. The
explanation follows.
In order for the points x1 >
x2 > ··· >
xn to be the “top n”
points of the sample process
at time t, we need (for j = 1,…,n) the
following: (i) an event of magnitude xj takes
place during the time period [0,t]; this occurs with
probability tλ(xj)
dxj; and (ii) no events of magnitude ∈
(xj,xj−1)
(where x0 is set to equal +∞) take place during
the time period [0,t]; this occurs with probability
exp{−t(Λ(xj) −
Λ(xj−1))}. Multiplying these
probabilities together yields the multidimensional density function
(21).
4. THE STRUCTURE OF THE SEQUENCE OF ORDER
STATISTICS
When viewed as a stochastic process in the “order
parameter” n (keeping the time t fixed), the
sequence of order statistics
{Xn(t)}n=1∞
conceals a hidden underlying structure, which we unveil in this section.
First, we reveal a Markovian structure governing the sequence of
order statistics. Second, we prove that the Markovian structure is due to
a hidden, underlying, Poissonian structure. This Poissonian
structure, in turn, provides us with a markedly simple simulation
algorithm for the sequence of order statistics. Third, we show that
the (discretely parameterized) sequence of order statistics can be
embedded in a simple transformation of a (continuously parameterized)
Gamma process.
4.1. Markovian Structure
The conditional distribution of a (n + 1)st order
statistic, given the value of the nth order statistic, is
(y < x). The explanation follows.
Given that the nth order statistic equals x, the
(n + 1)st order statistic will be less or equal to y
(where y < x) if and only if no events of magnitude
∈ (y,x) occur during the time interval
[0,t]; that is, if and only if
.
Since
is a Poisson point process, the random variable
is Poisson distributed with mean t(Λ(y) −
Λ(x)) and is independent of the points of
residing in [0,t] × [x,∞).
Hence, the left-hand side of (22) equals the probability that
,
which, in turn, is given by the right-hand side of (22).
Since (22) is a Markovian recursion, it implies that the
sequence of order statistics
{Xn(t)}n=1∞
is a Markov chain (in the variable n, for t
fixed). The initial condition of this Markov chain is given by the
distribution of the maximum X1(t); see
(14).
4.2. The Hidden Poissonian Structure
When viewed in the proper perspective, the sequence of order
statistics
{Xn(t)}n=1∞
conceals a hidden Poissonian structure. The “proper
perspective” is to consider the sequence
{Λ(Xn(t))}n=1∞,
rather than the original sequence
{Xn(t)}n=1∞.
Let us begin with the distribution of
Λ(X1(t)).
Using (14), we have
(u ≥ 0). Hence, Λ(X1(t))
is exponentially distributed with parameter t (mean
1/t). In a similar way, (19) implies that
Λ(Xn(t)) is Gamma distributed
with parameters (t,n).
More informative, however, is to analyze the conditional
distribution of the increment
Λ(Xn+1(t)) −
Λ(Xn(t)) given
Λ(Xn(t)). Indeed, using (22), we
have
that is, the increment
Λ(Xn+1(t)) −
Λ(Xn(t)) is independent
of Λ(Xn(t)) and is exponentially
distributed with parameter t (mean 1/t). Hence, we
have obtained the following proposition:
Proposition 1: Let
be an event process with characteristic Λ(x)
and order statistics
{Xn(t)}n=1∞.
Then the increasing sequence of points
{Λ(Xn(t))}n=1∞
forms a standard Poisson process with rate t.
4.3. Simulation and Gamma Embedding
Proposition 1 provides us with a remarkably simple simulation
algorithm for the entire sequence of order statistics
{Xn(t)}n=1∞:
where
{Zn}n=1∞
is an i.i.d. sequence of exponentially distributed random variables with
unit mean. In particular, the simulation algorithms for the
Fréchet, Weibull, and Gumbel order statistics are as follows:
Equation (23) is in fact a manifestation of a Gamma embedding
as we now explain.
The Gamma process (G(s))s≥0
is a stochastic process starting at the origin (G(0) = 0) whose
increments are independent, stationary, and Gamma
distributed:
(b > a ≥ 0). The Gamma process is a special
example of one-sided Lévy processes (Lévy subordinators)
[1].
At the point s = 1, the value of the Gamma process
G(1) is exponentially distributed with unit mean. Hence, since
the Gamma process has independent and stationary increments, the
increments {G(n) − G(n −
1)}n=1∞ form an i.i.d. sequence of
exponentially distributed random variables with unit mean. Combining this
observation with (23), we obtain the following Gamma embedding of the
sequence of order statistics
{Xn(t)}n=1∞:
that is, the discrete-parameter sequence
{Xn(t)}n=1∞
is embedded in the continuous-time process
(Λ−1(G(s)/t))s≥0,
which, in turn, is a transformation of the Gamma process
(G(s))s≥0.
Furthermore, the random variable
Λ−1(G(s)/t), for
noninteger parameter values s, may be considered a
“virtual” s-order statistic. Continuing yet another
step in this direction, we can define the Fréchet, Weibull, and
Gumbel virtual order processes as follows (s ≥
0):
5. THE INTERNAL HIERARCHY OF THE SEQUENCE OF
ORDER STATISTICS
What is the “internal hierarchy” of the sequence of the
order statistics? What are the relative magnitudes of the order
statistics? The answer to these questions is given by the following
proposition.
Proposition 2: Let
be an event process with characteristic Λ(x)
and order statistics
{Xn(t)}n=1∞.
Then the ratios
{Λ(Xn(t))n/Λ(Xn+1(t))n}n=1∞
are independent and uniformly distributed on the unit interval.
Equivalently,
The proof of Proposition 2, which is based on the use of an
order-preserving transformation of event processes, is provided in the
Appendix.
We note that the distribution of the ratio
Λ(Xn(t))/Λ(Xn+1(t))
can be deduced from Proposition 1. Indeed, (23) implies that this ratio
equals, in law, the ratio (Z1 +
··· +
Zn)/(Z1 +
··· + Zn+1), where
{Zn}n=1∞
is an i.i.d. sequence of exponentially distributed random variables with
unit mean. The latter ratio, in turn, is known (see, for example,
[6]) to be governed by the Beta
distribution function un (0 < u
< 1).
With Proposition 2 at hand, let us explore the internal hierarchy of
the sequence of order statistics in the special Fréchet, Weibull,
and Gumbel cases.
5.1. The Fréchet and Weibull Cases
In the Fréchet case, Λ(x) =
x−α, and in the Weibull case,
Λ(x) = |x|α (the
exponent α being a positive parameter). Hence, Proposition 2 implies
the following:
[bull ] Fréchet: The ratios
Xn+1(t)/Xn(t),
n = 1,2,…, are independent random variables
governed by the Beta distribution
[bull ] Weibull: The ratios
|Xn(t)|/|Xn+1(t)|,
n = 1,2,…, are independent random variables
governed by the Beta distribution
Most interesting is the first
ratio—X2(t)/X1(t)
in the Fréchet case and
|X1(t)|/|X2(t)|
in the Weibull case—which measures the relative magnitudes of the
“winner” and the “first runner-up.” The ratio
distribution, in both the Fréchet and Weibull cases, is governed by
the probability density function f (u) =
αuα−1 (0 < u < 1). This
density function undergoes a phase transition when crossing the parameter
value α = 1: (i) When 0 < α < 1, the density f
(u) is monotone decreasing, starting at f (0) = ∞
and decreasing to f (1) = α; (ii) when α = 1 (the
“critical” parameter value), the density f
(u) is uniform; and (iii) when α > 1, the density
f (u) is monotone increasing, starting at f (0)
= 0 and increasing to f (1) = α. Hence, the “first
runner-up” trails considerably behind the “winner” when
α is small, and it follows the “winner” closely when α
is large.
At the other end of the order spectrum, the distribution function
uαn becomes more and more concentrated
around the value u = 1 as n → ∞. This implies
that as n → ∞, the order statistics get closer and
closer to each other in ratio. In the Fréchet case, as
n → ∞, the order statistics converge to zero and hence
get closer and closer to each other, also absolutely. In the Weibull case,
on the other hand, the order statistics diverge to −∞ and
hence remain apart.
In the Fréchet and Weibull cases, Proposition 2 also provides
us with an algorithm for the simulation of the relative
contributions of the “top n” order statistics
X1(t),X2(t),…,Xn(t)
to their aggregate magnitude. The explanation follows.
Let
{Uk}k=1n−1
be an i.i.d. sequence of uniformly distributed random variables, and set
{Mk}k=0n−1
to be given by
Proposition 2 implies (all equalities below are equalities in
law):
Fréchet case:
Xk+1(t)/Xk(t)
= Uk1/kα and
hence, using recursion, Xk(t) =
X1(t)Mk−1.
This, in turn, yields the arithmetic average:
Weibull case:
|Xk(t)|/|Xk+1(t)|
= Uk1/kα and
hence, using recursion,
|Xk(t)| =
|Xn(t)|Mn−1
/Mk−1. This, in turn, yields the
harmonic average:
5.2. The Gumbel Case
In the Gumbel case Λ(x) = exp{−x} and
hence proposition 2 implies:
Gumbel: The differences
Xn(t) −
Xn+1(t), n = 1,2,…,
are independent random variables governed by the exponential
distribution
Alternatively, the random variables
{n(Xn(t) −
Xn+1(t))}n=1∞
are i.i.d. and exponentially distributed with unit mean.
Let us study the following “asymptotically centered”
sequence of differences
{Dn}n=1∞
defined by
Using (30), the sequence
{Dn}n=1∞
admits the probabilistic representation
where
{Zn}n=1∞
is an i.i.d. sequence of exponentially distributed random variables with
unit mean. The representation (31), in turn, yields that the mean and
variance of Dn are
where γ ≃ 0.577 is Euler's constant.
Moreover, the representation (31) further yields that the Laplace
transform of Dn is
(θ ≥ 0). Hence, using Stirling's formula, we have
The function Γ(1 + θ), however, is the Laplace transform of
the standard Gumbel distribution (see, for example, [5]), and hence we conclude the following:
The sequence
{Dn}n=1∞
converges, in law, to the standard Gumbel distribution.
6. RECORDS
Among the points of the event process
,
the set of record points (henceforth denoted by
) is of special interest. This last section is devoted to the
study of these points.
A point
is said to be a record point if and only if all events occurring
during the time period [0,t) were smaller in
magnitude than the value x:
If
,
then t is called a record time and x is called
a record value. We denote the sets of record times and record
values respectively by
.
Clearly, the sets
are the projections of the record set
on the time and space axes.
The probability that the points
{(tj,xj)}j=1n,
where t1 < t2 <
··· < tn and
x1 < x2 <
··· < xn, belong to
the record set
is given by
where t0 is set to equal zero. The explanation
follows.
In order for the points
{(tj,xj)}j=1n
to belong to the record set
,
we need the following (for j = 1,…,n): (i) the
point (tj,xj)
belongs to the underlying event process
;
this occurs with probability λ(xj)
dtj dxj; and
(ii) during the time interval
(tj−1,tj),
no events of magnitude greater or equal to the value
xj take place; this occurs with probability
exp{−(tj −
tj−1)Λ(xj)}.
Multiplying these probabilities together yields (33).
We note that (33) can be written, alternatively, in the form
where xn+1 is set to equal +∞.
Now, integrating (33) over x1 <
x2 < ··· <
xn yields
and integrating (34) over t1 <
t2 < ··· <
tn yields
Hence, using (5), we can conclude the following:
Proposition 3: Let
be an event process with characteristic Λ(x).
Then the following hold:
(i) The set of record times
is a temporal Poisson point process with rate
1/t.
(ii) The set of record values
a is Poisson point process (on
) with rate
λ(x)/Λ(x).
In particular, for the Fréchet, Weibull, and Gumbel cases the
rate λ(x)/Λ(x) of the Poisson point process
of record values
is as follows:
Several remarks are necessary:
(i) We emphasize that the record set itself
is not a Poisson point process on
since the probability
given in (34) does not admit the multiplicative form
(5).
(ii) Equation (35) is the continuous-time counterpart of
Rényi's record theorem [18]. Rényi's theorem asserts that if
{ξj}j=1n
is an i.i.d. sequence of random variables and
Ej is the event {ξj
is a record value}, then the events
{Ej}j=1n
are independent and P(Ej) =
1/j.
(iii) Results analogous to those given in (35) and (36) for
i.i.d. random sequences can be found in [3,
Chap.5].
(iv) Recall the waiting time L(t) defined
in Subsection 3.2: L(t) is the length of the time period
elapsing from time t until the first occurrence of a
record event after time t. Note that
{L(t) > l} if and only if the set of record
times
has no points on the interval
[t,t + l]. Hence, Proposition 3
yields that
7. CONCLUSIONS
Motivated by the fact that the classic EVT undertakes a discrete-time
approach, whereas most “real-world” systems are usually
continuous time, our aim in this work was to introduce a simple and
parsimonious model, counterpart to the discrete-time i.i.d. model, for the
study of extremes in continuous time.
To that end, we considered a generic continuous-time system in which
events of random magnitudes arrive stochastically—following
time-homogeneous Poisson point process dynamics on
.
The (monotone decreasing) function Λ(x) was used to denote
the Poissonian rate at which events of magnitude greater than x
occur and was assumed to satisfy
limx→−∞ Λ(x) = +∞
and limx→+∞ Λ(x) = 0. This
ensured the following: (i) the occurrences of the events are everywhere
dense on the time axis (i.e., there are countably many events occurring on
any time interval), whereas (ii) events greater than any given level occur
discretely (i.e., there are only finitely many such events on any time
interval).
This Poissonian model turned out to naturally accommodate the study
and investigation of extremes in continuous time:
- the maximum X1(t): the magnitude of the
greatest event occurring during the time interval
[0,t];
- the nth “runner-up”
Xn+1(t): the magnitude of the
(n + 1)st greatest event occurring during the time interval
[0,t];
- the “top n”
(X1(t),…,Xn(t)):
the vector of magnitudes of the n greatest events occurring
during the time interval [0,t];
- the set of record times Rtime: the time epochs
at which the record events occurred;
- the set of record values Rvalue: the values of
the record events.
Furthermore, for any fixed time t, we explored the sequence
of order statistics
{Xn(t)}n=1∞—viewed
as a stochastic process indexed by the parameter n:
- Structure: We discovered that the increasing sequence of
points
{Λ(Xn(t))}n=1∞
forms a standard Poisson Process with rate t.
- Simulation: We devised a markedly simple algorithm for the
simulation of the sequence
{Xn(t)}n=1∞.
- Hierarchy: We discovered that the ratios
{Λ(Xn(t))n/Λ(Xn+1(t))n}n=1∞
are independent and uniformly distributed on the unit interval.
Throughout, emphasis was focused on three special cases: (i)
Fréchet Λ(x) = x−α
(x > 0); (ii) Weibull Λ(x) =
|x|α (x < 0); (iii)
Gumbel Λ(x) = exp{−x} (x real). For
these cases, the resulting maximum distributions are governed respectively
by the Fréchet, Weibull, and Gumbel probability laws. These laws
are of major importance since they are the universal “stable
laws” of the classic EVT—the only possible maxima scaling
limits.
We hope that researchers in the engineering and informational
sciences, when having to deal with extreme events in continuous-time
systems, would find use in the modeling approach and results presented in
this work.
Acknowledgments
The author would like to thank the anonymous referee for his helpful
and constructive comments.
APPENDIX
We introduce an order-preserving transformation of event processes and
use it to prove Proposition 2.
A.1. An Order-Preserving Transformation
Given a point process
on [0,∞) × I and a monotone-increasing
bijection φ : I → J (I and J
being subintervals of the real line
), consider the transformation
that is, the point (t,x) of
is mapped by the transformation Φ to the point
(t,φ(x)) of
.
The map Φ transforms the point process
(on [0,∞) × I) to a new point process
(on [0,∞) × J). Furthermore, the map Φ
preserves the order of the order statistics: If
{Xn(t)}n=1∞
is the sequence of order statistics of the process
is the sequence of order statistics of the process
,
then
Now, if
is a Poisson point process with rate
,
then standard probabilistic arguments imply that
is also a Poisson point process and its rate is
.
In particular, if
is a sample process with characteristic
,
then
is a new sample process with characteristic
Equation (A.3) can also be derived directly by combining together
(A.2) and (14).
Hence, given a pair of sample processes—
with characteristic
with characteristic
—the map Φ induced by the monotone-increasing
bijection
transforms the process
to the process
.
Equations (A.3) and (A.4) enable us to transform between
different sample processes while preserving the order of their
order statistics.
A.2. Proof of Proposition 2
We split the proof into two steps. In the first step, we prove that
Proposition 2 holds for the special case of an event process
with characteristic
.
In the second step, we use the order-preserving transformation of
Subsection A.1 to validate Proposition 2 for any event process
.
We remark that the running maximum process
(Y1(t))t≥0 (associated
with the event process
) is analogous, albeit not identical, to the “Poisson
hyperbolic staircase process” introduced and studied in [16].
Step 1: Fix an integer n. The characteristic of the
event process
is
and, hence, (21) implies that the joint probability density function of
the vector of order statistics
(Y1(t),…,Yn+1(t))
is given by
Now,
This, in turn, implies that ratios
{Yj+1(t)/Yj(t)}j=1n
are independent random variables governed by the Beta
distribution:
Step 2: Let
be an arbitrary event process with characteristic
and let
be an event process with characteristic
.
Equation (A.4) implies that the order-preserving map Φ transforming
the process
to the process
is induced by the monotone-increasing bijection
(or, equivalently, using (A.3): if
,
then
.
Exploiting the fact that Φ is order-preserving, (A.2) implies
that
Hence, using Step 1, we obtain that the ratios
are independent random variables governed by the Beta
distribution:
Since the choice of n was arbitrary, the proof of Proposition
2 is complete.
