1 The setting
We discuss the probabilistic background for the diffusion (5) and specify the assumptions which will in force throughout the paper. Assumptions and results are stated in a first subsection, proofs and additional remarks in a second one.
1.1 Assumptions, null recurrence, likelihoods, some estimators
Throughout this paper, we consider the model (5), with and the associated parameter given by
The functions in equation (5) satisfy the following.
Assumption 1: The functions in (5) are Lipschitz continuous and such that
i) finite limits do exist for as , and are denoted by ;
ii) for arbitrarily small, ;
iii) the functions occurring in (5) are linearly independent in the following sense:
for open in and real constants , any representation on implies .
Note that the functions or considered in example 1 above do satisfy all requirements of assumption 1: iii) holds in both cases or , ii) is obvious, and elementary arguments (such as for all from [He 93] Ch. 87) allow to check i). We put
on . Here , thus null recurrence holds for every .
We strengthen that the Lebesgue density of the invariant measure varies regularly as and as . The index of regular variation , the same on the left and the right branch, depends on only and ranges over the interval . We have no regular variation for the functions in the drift, cf. example (6), whereas they contribute to asymptotic constants in virtue of assumption 1 i). We have choosen as the maximal open interval on which null recurrence holds.
Throughout this paper, the starting point for the diffusion (5) does not depend on and will be fixed. From now on, these assumptions and notations will remain in force (and we omit to recall this in the results below).
For the theoretical background of what follows we refer to the classical books Liptser and Shiryaev [LS 78] and Jacod and Shiryaev [JS 87]; see also Ibragimov and Khasminskii [IH 81], Kutoyants [Ku 04], or section 6.2 in Höpfner [Ho 14]. Let denote the law of under on the canonical path space for cadlag processes ([JS 87] Chapter VI), with canonical filtration
Write for the canonical process, i.e. the process of coordinate projections , , and for the -martingale part of . We introduce a -dimensional -martingale :
where denotes the -martingale part of . will be called score martingale. The angle bracket of under is the process
(with ranging over ) which is observable; will be called information process.
Lemma 2: For all pairs in , the laws are locally equivalent relative to . The log-likelihood ratio process of with respect to relative to is given by
where is the scalar product in . The information , , takes values in the set of all strictly positive definite symmetric matrices, -almost surely for every .
There are many possibilities to define estimators for the unknown parameter based on time-continuous observation of the diffusion path (5) on .
Proposition 1: We have well defined maximum likelihood (ML) estimators, of form
where , denotes common determinations
|for the family of stochastic integrals under|
|for the family of stochastic integrals under|
valid jointly under all . For every , the ML estimation error is such that
One might prefer to restrict observation of a null recurrent process to some fixed (and sufficiently large) compact set in , and thus consider estimators of the following type.
Proposition 2: Fix compact in such that the starting point of the diffusion is an interior point of . Define , , in analogy to (9), (10) and above, but with , replaced by , . Then for takes values in , -almost surely for every , and the estimator
admits a representation
for every . Under , this estimator replaces the log-likelihood surface considered in (11) by a modified (again ’inverse bowl-shaped’) surface
as an estimator for . In our model (5), this estimator is inconsistent: it admits a represention
under , with additional terms which by the ratio limit theorem
converge -almost surely as .
1.2 The proofs
Proof of lemma 1: The proof uses classical arguments, see e.g. Gihman and Skorohod [GS 72] or Khasminskii [Kh 80]; a resumé is in [Ho 14] sections 9.2–9.4. Fix , write for short for the drift in equation (5) under
and define functions , on by
where , and we deduce from assumption 1 that
is a bijection onto . As a consequence, the diffusion (5) is recurrent under with invariant measure
by proposition 9.12 a) in [Ho 14].
Remark 2: in (7) is a maximal open parameter interval such that diffusion (5) is null recurrent:
The explicit representation (15) of , valid with , shows that is integrable whenever (i.e. ): then in (14) converges to finite limits as , thus is transient by [GS 72], lemma 3 on p. 117.
In the limiting case () to transience, in (14), of logarithmic growth as , is a bijection onto . Thus all cases () lead to recurrence of with invariant measure , cf. proposition 9.12 a) in [Ho 14]. Ergodicity (positive recurrence) holds if the invariant measure is a finite measure, i.e. for ().
Proof of lemma 2: See [LS 78] and [JS 87] for local absolute continuity and structure of likelihood ratio processes, a short resumé is 6.9–6.12 in [Ho 14], whence the representation (11) for log-likelihood ratio processes. Assumption 1 implies that all are bounded functions, thus local -martingales defined by (9) are square-integrable martingales with angle bracket (10) under . It remains to prove that -almost surely, the process takes values in , the set of strictly positive definite symmetric matrices in . For with ,
where we write for short
for the row vector with entries. Now -almost surely, the range of the diffusion path (5) has non-empty interior; on this (random, since defined by the path) open set, the functions are linearly independent by assumption 1. Hence -almost surely, the right hand side in (18) is strictly positive, first pointwise in and , second uniformly on by continuity, and we let .
Proof of proposition 1: Common determinations of the stochastic integrals jointly for all
do exist since probability measures, , are locally equivalent relative to , see lemma 8.2’ in [Ho 14]. Now we exploit almost sure invertibility of the information at time combined with ’inverse bowl shape’ of the log-likelihoods.
Proof of proposition 2: By assumption, the starting point is an interior point of . Thus for , almost surely contains open balls, and (18) remains true with in place of .
Proof for remark 1: corresponds to the maximum of the one-dimensional surface
This is valid for all and for arbitrary choice of a starting point for the process (5).
We formulate a theorem on convergence of additive functionals and martingale additive functionals in the null recurrent diffusion (5). It combines theorem 3.1 from Höpfner and Löcherbach [HL 03] with results due to Khasminskii ([Kh 80], or theorem 2.2 in [KY 00] and theorem 1.1 in [Kh 01]). The approach is analoguous to [HK 03] or to examples 3.5 and 3.10 in [HL 03].
2.1 Convergence of martingales together with their angle bracket
Introducing further notation for , we define
(cf. (8), assumption 1 and lemma 1) and introduce a -valued index
together with weights
Functions which satisfy (23) do belong to for every , by lemma 1, and we can define
Then for every , we have weak convergence in the Skorohod space of
with notation (24). Here is a Mittag-Leffler process of index , the process inverse (i.e. the process of level crossing times) of a stable increasing process with index , and a -dimensional Brownian motion which is independent of ; thus is Brownian motion subject to independent time change .
The one-dimensional case of theorem 1 corresponds to theorem A in [HK 03] (note that the invariant measure there has a factor with respect to our in lemma 1).
Remark 3: The one-sided stable process with index has stationary independent increments with Laplace transform
Its process inverse has continuous nondecreasing paths with and , hence , are continuous processes. As a consequence ([JS 87], VI.2.3) of the continuous mapping theorem, theorem 1 implies weak convergence in of (26) evaluated at time to (27) evaluated at time .
Remark 4: is concentrated on . Whereas
admits finite moments of arbitrary order, there is no finite moment of order : we have
which will be of importance below.
2.2 Proof of theorem 1
Lemma 3: The path of can be decomposed into iid life cycles , , defined by
(up to an initial segment ), with the function inverse of in (16) which depends on . For this decomposition the following holds:
i) for nonneative and measurable,
Proof: 1) The function in (14) is harmonic for the Markov generator of the diffusion . In a first step we consider transformed by : is a diffusion without drift
(with constant in (5), and is the function inverse of ) whose invariant measure is given by
This is the Lebesgue density of the invariant measure in (29) for the process .
2) We comment on the invariant density for . Multiplying the asymptotic constant by we define
takes values in and is such that
define iid life cycles in the path of for , with the following two properties: first,
for measurable functions and as in (29); second,
4) The iid life cycles for in step 3) are iid life cycles for
Using step 2), the asymptotic constant on the right hand side of (33) equals
and lemma 3 is proved by step 3); recall that both and depend on .
Proof of theorem 1: 1) In a first step, consider convergence of martingales in case .
Fix and apply theorem 3.1 c) of [HL 03] to one-dimensional measurable functions which belong to : to the sequence of renewal times considered in lemma 3 correspond norming sequences
converges weakly as in the Skorohod space of one-dimensional cadlag functions to
where is given by
here we use (32) together with . Taking into account this factor arising on the right hand side of (34) we arrive at the norming sequence of theorem 1, defined by (25) and (22), and have weak convergence under
with as asserted in theorem 1. This proves convergence of martingales in case .
2) The result proved in step 1) can be extended to case and to weak convergence of martingales together with their angle bracket: apply corollaries 3.2 and 3.3 in [HL 03].
Proof for remark 3: We refer to 2.5-2.8 in [HL 03] and the references quoted there.
Proof for remark 4: It is well known that is finite for and infinite for (e.g. [Ho 14], 6.18’). Since is the process inverse to , the relation
holds and proves that equals .
3 LAMN and optimal estimator sequences
3.1 Limit distribution for the estimators of section 1.1
for all , where is -dimensional Brownian motion independent of the Mittag-Leffler variable , and where is given by
with , ranging from to . If we fix as in proposition 2 a compact interval such that and replace , by ,