A general framework for perfect simulation of long memory processes

Abstract

In this paper a general approach for the perfect simulation of a stationary process with at most countable state space is outlined. The process is specified through a kernel, prescribing the probability of each state conditional to the whole past history. We follow the seminal paper of Comets, Fernandez and Ferrari, where sufficient conditions for the construction of a certain perfect simulation algorithm have been given. We generalize this approach by defining backward coalescence times for these kind of processes; this allows us to construct perfect simulation algorithms under weaker conditions. Comment: 25 pages

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arXiv:1004.0113v1 [math.PR] 1 Apr 2010
A general framework for perfect simulation of long
memory processes
Emilio De Santis
Dipartimento di Matematica
Sapienza Universit` a di Roma
desantis@mat.uniroma1.it
Mauro Piccioni
Dipartimento di Matematica
Sapienza Universit` a di Roma
piccioni@mat.uniroma1.it
April 2, 2010
Abstract
In this paper a general approach for the perfect simulation of a stationary process
with at most countable state space is outlined. The process is specified through a
kernel, prescribing the probability of each state conditional to the whole past history.
We follow the seminal paper [CFF02], where sufficient conditions for the construction
of a certain perfect simulation algorithm have been given. We generalize this approach
by defining backward coalescence times for these kind of processes; this allows us to
construct perfect simulation algorithms under weaker conditions. Backward coales-
cence times are constructed in the following ways: (i) by taking into account some
a priori knowledge about the histories that occur; (ii) by merging the algorithm in
[CFF02] with the classical CFTP algorithm [PW96].
Keywords: Perfect simulation, Coupling, Renewal processes.
AMS classification: 60G10, 60J10.
1Introduction
Perfect simulation algorithms for stochastic processes have been developed mostly for Markov
chains, starting from the original CFTP algorithm presented in the founding paper by Propp
and Wilson [PW96]. Later on, Foss and Tweedie [FT98] recognized the fundamental role
of the so-called stochastic recursive sequences for perfect simulation. Murdoch and Green
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[MG98] constructed a stochastic recursive sequence for perfect simulation when the transi-
tion kernel satisfies a minorization condition, called the gamma-coupler. Among the appli-
cations of perfect simulation in recent years, we cite stochastic geometry ([FFG02], [Møl01])
and random fields ([HS00], [DSP08]).
The fact that the main idea underlying stochastic recursive sequences, and the gamma-
coupler in particular, works beyond the markovian case, is shown by the extension, due to
Comets et al [CFF02], to processes with infinite memory. The aim of the present work is to
present some generalizations of their results.
In this paper we consider stochastic processes defined on Z with values in an alphabet G,
which is finite or countable: thus realizations of these processes are two-sided infinite words.
The law of the process is obtained through a transition kernel prescribing the probability
that each letter of the alphabet occurs in any given position of the word, conditional to the
whole history preceding it.
For notational convenience we use the following convention concerning sequences with
values in G: whenever m ≤ n are elements of Z we define the word
sn
m= (sn,...,sm) ∈ Gm−n+1.
With an obvious extension we also allow m = −∞ and n = +∞. For brevity of notation we
write s := s+∞
elements of G−N∗, where N∗is the set of positive integers. We can concatenate any word
sn
−∞, obtaining another history
−∞. For m = −∞ and n finite the word will be called a history. Histories are
mwith a history wm−1
(sn
m,wm−1
−∞) = (sn,...,sm,wm−1,wm−2,...).
The same notational conventions are used for sequences with values in the interval [0,1).
The set of histories G−N∗is equipped with the ultrametric distance
δ(s−1
−∞,t−1
−∞) = 2−inf{n:s−n?=t−n}, s−1
−∞,t−1
−∞∈ G−N∗.
The corresponding Borel σ-algebra coincides with the product σ-algebra, which is generated
by all cylinder sets.
Let us consider a kernel p : G × G−N∗→ [0,1], which will be denoted by p(g|w−1
This means that, for any g ∈ G, p(g|·) is a measurable function in G−N∗such that
?
g ∈ G
P (Xm= g|Xm−i,i ∈ N∗) = p(g|Xm−1
For any n ∈ N+, the ”one-dimensional” kernel p induces an (n + 1)-dimensional kernel
−∞).
g∈Gp(g|w−1
We say that a process {Xn,n ∈ Z} is compatible with the kernel p, if for any m ∈ Z and
−∞) = 1.
−∞), a.s. (1)
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p(n+1): Gn+1× G−N+→ [0,1] defined by
p(n+1)(gn,...,g0|w−1
−∞) =
n?
k=0
p(gk|gk−1,...,g0,w−1
−∞), gk∈ G, k = 0,1,...,n, w−1
−∞∈ G−N+.
(2)
If the process {Xn,n ∈ Z} is compatible with p, then for any m ∈ Z, n ∈ N and any choice
of gk∈ G,k = 0,...,n
P (Xm+k= gk,k = 0,...,n|Xm−i,i ∈ N∗) =
n
?
k=0
p(gk|gk−1,...,g0,Xm−1
−∞) a.s. (3)
Starting from a kernel p, perfect simulation is aimed to construct algorithms for sampling
a compatible process {Xn,n ∈ Z}, giving at the same time sufficient conditions for its
uniqueness.
Processes of this type are known in the literature as random systems with complete
connections: for the foundations of their theory see [IG90], whereas for a full account about
the literature on these processes we refer to the introduction in [CFF02]. In particular,
previously known uniqueness conditions were given in [Ber87] and [Lal00].
We conclude this introduction by giving a plan of the paper. In Section 2 we introduce
general coupling functions in the context of processes with infinite memory and define
a general backward coalescence time. Its existence allows to deduce perfect simulation
algorithms for the unique stationary process compatible with the kernel. At the end of the
section the ”maximal” coupling function is described, as introduced in [CFF02].
From the nature of the backward coalescence time used in [CFF02] we abstract the notion
of information depth, to which Section 3 is devoted. This is a stopping time, associated
to each instant and adapted to the past values of the random sources feeding the coupling
function, which bounds the amount of information needed on the previous states in order to
compute the state at that instant. From an information depth we give a canonical way to
construct a backward coalescence time, provided it can be shown that it is a.s. finite. As in
[CFF02], under slightly stronger condition, there is also a regeneration structure, which can
be exploited to simulate the process for all positive times and not just on a finite window.
In Section 4 we modify the information depth used in [CFF02] in order to deal with
examples like alternating renewal process and more general processes with a random change
of regime in the memory of the past, for which the algorithm in [CFF02] is not successful.
The main ingredient for proving that our modification works is the knowledge of the histories
that could occur, once the sources of randomness are produced backward in time.
Finally, in Section 5 we construct directly a backward coalescence time, when the in-
formation depth in [CFF02] is always strictly positive, denying the possibility of building
from it a backward coalescence time. In general, the construction of this modified backward
coalescence times needs the modification of the coupling function as well. The construction
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also requires a positive probability of coalescence in what we call the markovian regime:
under suitable conditions, a perfect simulation algorithm is constructed by combining the
algorithm in [CFF02] with the classical CFTP in [PW96]. A class of examples in which
these conditions hold is finally discussed.
2 Coupling functions, backward coalescence times and
perfect simulation
In this section we discuss some general issues involved in the design of a perfect simulation
algorithm for a process compatible with a kernel p : G×G−N∗→ [0,1] of the form described
above. The first concept to be introduced is that of coupling function. Despite the fact that
in all the examples presented throughout the paper the coupling function is almost always
the same, borrowed from [CFF02], we believe that it is useful to give an abstract definition.
In particular we choose to make explicit the dependence of the backward coalescence time,
which is defined afterwards, on the coupling function. In fact, several backward coalescence
times will be discussed throughout the paper, different from the one used in [CFF02], for
the same coupling function.
We first give the definition of admissible history, which is related to the zeros of the
kernel function. This concept will be useful in the definitions and the results that follow.
We start with the definition of a forbidden word of the alphabet G, recursively on the
length. A letter g of the alphabet G is forbidden if p(g|w−1
word s0
−∞) = 0, for any w−1
−∞∈ G−N∗. A
−nis forbidden or
−n= (s0,...,s−n+1,s−n) of length n + 1 is forbidden if either s−1
p(s0|w−1
−∞) = 0, ∀w−1
−∞∈ G−N∗: w−1
−n= s−1
−n.
For any n ∈ N+, we call Hnthe set of words that are not forbidden, of length n.
Next define the set of admissible histories
H = {w−1
−∞∈ G−N∗such that w−1
−∞∈ G−N+: w−1
−n∈ Hn,n ∈ N+}.
−n= s−1
Since the set of histories w−1
closed, the set H, being an intersection of sets of this form, is closed.
Now we prove that a process X, which is compatible with p, is such that X0
H with probability 1. It is readily seen that it is enough to prove that if s0
word for p, then
P(X0= s0,...,X−n= s−n) = 0.
−nfor any fixed s−1
−n∈ Gn+1is
−∞belongs to
−nis a forbidden
(4)
In order to prove this, notice that by the definition of forbidden word and (2) it is
p(n+1)(s0,...,s−n|w−n−1
−∞) = 0,
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for any w−n−1
we conclude that (4) holds. With exactly the same argument it is proved that Xm
to H w.p. 1 as well, for any m ∈ Z.
In principle, the construction of the set of admissible histories can be iterated, replacing
in the above definitions G−N+with H. In this way the set of forbidden words could be
enlarged, and thus the set of admissible histories could be reduced, and so on. We choose
not to pursue this kind of generalization, since the previous definition is adequate for the
examples which will be presented during the paper.
A coupling function f for the kernel p is a function f : [0,1) × H → G such that
for any w−1
union of intervals [ci(g|w−1
simulation purposes we also assume that ci(g|w−1
looking at a finite portion of the history w−1
and for any u ∈ [0,1) the function f(u|·) is continuous in H.
Since an interval [c,d) is either empty or it has positive length, the set of admissible
histories is invariant under the coupling function, in the sense that for any u ∈ [0,1)
w−1
−∞
∈ G−N∗. By integrating over w−n−1
−∞
with respect to the law of the process
−∞belongs
−∞∈ H and any g ∈ G the set {u ∈ [0,1) : f(u|w−1
−∞),di(g|w−1
−∞) = g} is a disjoint
−∞
?. For practical
−∞)), i ∈ N+, of total length p?g|w−1
−∞. This implies that f is jointly measurable
−∞) and di(g|w−1
−∞) can be computed by
−∞∈ H ⇒ (f(u|w−1
−∞),w−1
−∞) ∈ H.
As a consequence of the definition, if f is a coupling function for p and U is a random
variable uniformly distributed in [0,1), then
P(f(U|w−1
−∞) = g) = p?g|w−1
−∞
?, for each g ∈ G, w−1
−∞∈ H.(5)
Starting from f(1)= f, we define recursively f(n): [0,1)n× H → G for any n ∈ N, in
the following natural way
f(n+1)(un,...,u0|w−1
Thus, whenever Uk,k = 0,...,n are i.i.d. random variables with the uniform distribution
in [0,1), the random vector?f(k+1)(Uk,...,U0|w−1
possible to define, in the same probability space, a family of processes evolving in forward
time according to the given kernel p, indexed by all admissible histories w−1
For the implementation of a perfect simulation algorithm we require a coupling function
f to admit a backward coalescence time, which we are going to define.
Consider a sequence U ={Ui,i ∈ Z} of i.i.d. random variables, with uniform distribution
in the interval [0,1), used as the source of randomness for the construction of the processes
of interest. For any m,n ∈ Z with m ≤ n we define the σ-algebra Fn
For simplicity of notation we specify an arbitrary reference admissible history g−1
say that a measurable function τ0(U0
time if it has the properties:
−∞) := f?un|f(n)?un−1,...,u0|w−1
−∞
?,...,f(1)(u0|w−1
−∞),w−1
−∞
?. (6)
−∞),k = 0,...,n?has the law p(n+1)given in
(2), for any w−1
−∞∈ H. This means that through the iterations of the coupling function it is
−∞∈ H.
m= σ (Ui,i = m,...,n).
−∞∈ H. We
−∞) with non positive values is a backward coalescence
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