LAMN in a class of parametric models for null recurrent diffusion

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 $f_{1}$ and the associated parameter given by

(7)

f_{1} (x) = \frac{x}{1 + x^{2}}, ϑ_{1} \in Θ_{1}, Θ_{1} := (- \frac{1}{2} σ^{2}, + \frac{1}{2} σ^{2}) .

The functions $f_{2, 1}, \dots, f_{2, m}$ in equation (5) satisfy the following.

Assumption 1: The functions $f_{2, ν} : I R \to I R$ in (5) are Lipschitz continuous and such that
i) finite limits do exist for $F_{2, ν} (x) := \int_{0}^{x} f_{2, ν} (y) d y$ as $x \to \pm \infty$ , and are denoted by $F_{2, ν} (\pm \infty) \in I R$ ;
ii) for $ε > 0$ arbitrarily small, $\int_{{| x | > 1}} [f_{2, ν}]^{2} (x) | x |^{1 - ε} d x < \infty$ ;
iii) the functions occurring in (5) are linearly independent in the following sense:
for $U$ open in $I R$ and real constants $ζ_{1}, ζ_{2, 1}, \dots, ζ_{2, m}$ , any representation $0 \equiv ζ_{1} f_{1} + m \sum ν = 1 ζ_{2, ν} f_{2, ν}$ on $U$ implies $0 = ζ_{1} = ζ_{2, 1} = \dots = ζ_{2, m}$ .

Note that the functions $f_{2}$ or $f_{2, 1}, \dots, f_{2, 2 ℓ}$ considered in example 1 above do satisfy all requirements of assumption 1: iii) holds in both cases $m = 1$ or $m = 2 ℓ$ , ii) is obvious, and elementary arguments (such as $| \int_{s}^{t} \frac{sin y}{y} d y | < \frac{2}{s}$ for all $0 < s < t < \infty$ from [He 93] Ch. 87) allow to check i). We put

(8)

Θ := Θ_{1} \times I R^{m}, ϑ = (ϑ_{1}, ϑ_{2, 1}, \dots, ϑ_{2, m}), λ_{1} (ϑ) := \frac{2}{σ^{2}} ϑ_{1}, λ_{2, ν} (ϑ) := \frac{2}{σ^{2}} ϑ_{2, ν} .

Lemma 1: Under assumption 1, the diffusion (5) is recurrent in the sense of Harris for every $ϑ \in Θ$ . With notation (8), the invariant measure of $ξ = (ξ_{t})_{t \geq 0}$ , unique up to constant multiples, is given by

μ^{ϑ} (d x) = \frac{1}{σ^{2}} (\sqrt{1 + x^{2}})^{λ_{1} (ϑ)} exp {m \sum ν = 1 λ_{2, ν} (ϑ) F_{2, ν} (x)} d x, ϑ \in Θ

on $(I R, B (I R))$ . Here $μ^{ϑ} (I R) = \infty$ , thus null recurrence holds for every $ϑ \in Θ$ .

We strengthen that the Lebesgue density of the invariant measure $μ^{ϑ}$ varies regularly as $x \to + \infty$ and as $x \to - \infty$ . The index of regular variation $λ_{1} (ϑ)$ , the same on the left and the right branch, depends on $ϑ_{1}$ only and ranges over the interval $(- 1, 1)$ . We have no regular variation for the functions $f_{2, ν}$ in the drift, cf. example (6), whereas they contribute to asymptotic constants in virtue of assumption 1 i). We have choosen $Θ_{1}$ as the maximal open interval on which null recurrence holds.

Throughout this paper, the starting point $x_{0} \in I R$ for the diffusion (5) does not depend on $ϑ \in Θ$ 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 $Q^{ϑ}$ denote the law of $ξ = (ξ_{t})_{t \geq 0}$ under $ϑ$ on the canonical path space $(D, D, I F)$ for cadlag processes ([JS 87] Chapter VI), with canonical filtration

I F = (F_{t})_{t \geq 0}, F_{t} = ⋂ r > t σ (η_{s} : 0 \leq s \leq r) .

Write $η = (η_{t})_{t \geq 0}$ for the canonical process, i.e. the process of coordinate projections $η_{t} (f) = f (t)$ , $f \in D$ , and $m^{(ϑ)}$ for the $Q^{ϑ}$ -martingale part of $η$ . We introduce a $(1 + m)$ -dimensional $Q^{ϑ}$ -martingale $S (ϑ) = {(S (ϑ)_{t})}_{t \geq 0}$ :

(9)

where $m^{(ϑ)}$ denotes the $Q^{ϑ}$ -martingale part of $η$ . $S (ϑ)$ will be called score martingale. The angle bracket $⟨ S (ϑ) ⟩$ of $S (ϑ)$ under $Q^{ϑ}$ is the process $J = (J_{t})_{t \geq 0}$

(10)

J_{t} = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \begin{matrix} \int_{0}^{t} \frac{1}{σ^{2}} [f_{1}]^{2} (η_{s}) d s & \dots & \int_{0}^{t} \frac{1}{σ^{2}} [f_{1} f_{2, ν^{'}}] (η_{s}) d s & \dots ⋮ & ⋱ \int_{0}^{t} \frac{1}{σ^{2}} [f_{2, ν} f_{1}] (η_{s}) d s & \int_{0}^{t} \frac{1}{σ^{2}} [f_{2, ν} f_{2, ν^{'}}] (η_{s}) d s ⋮ & ⋱ \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠, t \geq 0

(with $ν, ν^{'}$ ranging over $1, \dots, m$ ) which is observable; $J$ will be called information process.

Lemma 2: For all pairs $ϑ^{'}, ϑ$ in $Θ$ , the laws $Q^{ϑ^{'}}, Q^{ϑ}$ are locally equivalent relative to $I F$ . The log-likelihood ratio process of $Q^{ϑ^{'}}$ with respect to $Q^{ϑ}$ relative to $I F$ is given by

(11)

Λ_{t}^{ϑ^{'} / ϑ} := (ϑ^{'} - ϑ)^{⊤} S (ϑ)_{t} - \frac{1}{2} (ϑ^{'} - ϑ)^{⊤} J_{t} (ϑ^{'} - ϑ), t \geq 0

where $\cdot \cdot^{⊤}$ is the scalar product in $I R^{1 + m}$ . The information $J_{t}$ , $0 < t < \infty$ , takes values in the set $D^{+}$ of all strictly positive definite symmetric $(1 + m) \times (1 + m)$ matrices, $Q^{ϑ}$ -almost surely for every $ϑ \in Θ$ .

There are many possibilities to define estimators for the unknown parameter $ϑ \in Θ$ based on time-continuous observation of the diffusion path (5) on $[0, t]$ .

Proposition 1: We have well defined maximum likelihood (ML) estimators, of form

{ˆ ϑ}_{t} = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \begin{matrix} {ˆ ϑ}_{t}^{(1)} {ˆ ϑ}_{t}^{(2, 1)} ⋮ {ˆ ϑ}_{t}^{(2, m)} \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ := 1_{{J_{t} \in D^{+}}} {[J_{t}]}_{t}^{- 1} Y_{t}, Y_{t} = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \begin{matrix} Y_{t}^{(1)} Y_{t}^{(2, 1)} ⋮ Y_{t}^{(2, m)} \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠, 0 < t < \infty

where $Y^{(1)}$ , $Y^{(2, ν)}$ denotes common determinations

		$(t, ω) \to Y^{(1)} (t, ω)$	for the family of stochastic integrals $\int \frac{1}{σ^{2}} f_{1} (η_{s}) d η_{s}$ under $Q^{ϑ}$
		$(t, ω) \to Y^{(2, ν)} (t, ω)$	for the family of stochastic integrals $\int_{0}^{t} \frac{1}{σ^{2}} f_{2, ν} (η_{s}) d η_{s}$ under $Q^{ϑ}$

valid jointly under all $ϑ \in Θ$ . For every $ϑ \in Θ$ , the ML estimation error is such that

(12)

{ˆ ϑ}_{t} - ϑ = 1_{{J_{t} \in D^{+}}} [J_{t}]^{- 1} S (ϑ)_{t} Q^{ϑ} -almost surely% .

One might prefer to restrict observation of a null recurrent process to some fixed (and sufficiently large) compact set $A$ in $I R$ , and thus consider estimators of the following type.

Proposition 2: Fix $A$ compact in $I R$ such that the starting point $x_{0}$ of the diffusion is an interior point of $A$ . Define $S (ϑ, A)$ , $J (A)$ , $Y (A)$ in analogy to (9), (10) and $Y$ above, but with $f_{1}$ , $f_{2, 1}, \dots, f_{2, m}$ replaced by $f_{1} \cdot 1_{A}$ , $f_{2, 1} \cdot 1_{A}, \dots, f_{2, m} \cdot 1_{A}$ . Then $J (A)_{t}$ for $0 < t < \infty$ takes values in $D^{+}$ , $Q^{ϑ}$ -almost surely for every $ϑ \in Θ$ , and the estimator

˜ ϑ (A)_{t} := 1_{{J (A)_{t} \in D^{+}}} {[J (A)_{t}]}_{t}^{- 1} Y (A)_{t}

admits a representation

(13)

˜ ϑ (A)_{t} - ϑ = 1_{{J (A)_{t} \in D^{+}}} [J (A)_{t}]^{- 1} S (ϑ, A)_{t} Q^{ϑ} -almost% surely

for every $ϑ \in Θ$ . Under $ϑ \in Θ$ , this estimator replaces the log-likelihood surface $ϑ^{'} \to Λ_{t}^{ϑ^{'} / ϑ}$ considered in (11) by a modified (again ’inverse bowl-shaped’) surface

Θ ∋ ϑ^{'} ⟶ (ϑ^{'} - ϑ)^{⊤} S (ϑ, A)_{t} - \frac{1}{2} (ϑ^{'} - ϑ)^{⊤} J (A)_{t} (ϑ^{'} - ϑ) .

For the principal parameter $ϑ_{1} \in Θ_{1}$ we might think of an estimator which was optimal in the one-dimensional model (1) considered in section 2 of [HK 03].

Remark 1: Write ${ˇ J}_{t} = \int_{0}^{t} \frac{1}{σ^{2}} [f_{1}]^{2} (η_{s}) d s$ for the $(1, 1)$ -entry of $J_{t}$ in (10), $ˇ S (ϑ)_{t}$ for the first component of $S (ϑ)_{t}$ in (9), and ${ˇ Y}_{t}$ for the first component of $Y_{t}$ defined above. Consider

ˇ ϑ (t) := [{ˇ J}_{t}]^{- 1} {ˇ Y}_{t} = \frac{\int_{0}^{t} f_{1} (η_{s}) d η_{s}}{\int_{0}^{t} [f_{1}]^{2} (η_{s}) d s}, 0 < t < \infty

as an estimator for $ϑ_{1} \in Θ_{1}$ . In our model (5), this estimator is inconsistent: it admits a represention

ˇ ϑ (t) - ϑ_{1} = {ˇ b}_{t} (ϑ) + [{ˇ J}_{t}]^{- 1} ˇ S (ϑ)_{t}

under $Q^{ϑ}$ , with additional terms ${ˇ b}_{t} (ϑ)$ which by the ratio limit theorem

{ˇ b}_{t} (ϑ) = m \sum ν = 1 ϑ_{2, ν} \frac{\int_{0}^{t} [f_{1} f_{2, ν}] (η_{s}) d s}{\int_{0}^{t} [f_{1}]^{2} (η_{s}) d s} ⟶ m \sum ν = 1 ϑ_{2, ν} \frac{μ^{ϑ} (f_{1} f_{2, ν})}{μ^{ϑ} (f_{1}^{2})} % a s $ t \to \infty $

converge $Q^{ϑ}$ -almost surely as $t \to \infty$ .

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 $ϑ$ , write for short $b$ for the drift in equation (5) under $ϑ$

b (x) = (ϑ_{1} f_{1} + m \sum ν = 1 ϑ_{2, ν} f_{2, ν}) (x), x \in I R,

and define functions $s$ , $S$ on $I R$ by

(14)

S (x) := \int_{0}^{x} s (y) d y, s (y) := exp (- \int_{0}^{y} \frac{2 b}{} σ^{2} (v) d v) .

(7) and (8) together yield

(15)

s (y) = {(\sqrt{1 + y^{2}})}^{- λ_{1} (ϑ)} exp (- m \sum ν = 1 λ_{2, ν} (ϑ) F_{2, ν} (y)), y \in I R

where $- 1 < λ_{1} (ϑ) < 1$ , and we deduce from assumption 1 that

(16)

S (x) \sim sgn (x) \frac{1}{1 - λ_{1} (ϑ)} | x |^{1 - λ_{1} (ϑ)} exp (- m \sum ν = 1 λ_{2, ν} (ϑ) F_{2, ν} (\pm \infty)) as x \to \pm \infty

is a bijection onto $I R$ . As a consequence, the diffusion (5) is recurrent under $ϑ$ with invariant measure

(17)

μ^{ϑ} (d y) = \frac{1}{σ^{2}} \frac{1}{s (y)} d y = \frac{1}{σ^{2}} {(\sqrt{1 + y^{2}})}^{+ λ_{1} (ϑ)} exp (+ m \sum ν = 1 λ_{2, ν} (ϑ) F_{2, ν} (y))

by proposition 9.12 a) in [Ho 14]. $□$

Remark 2: $Θ_{1}$ in (7) is a maximal open parameter interval such that diffusion (5) is null recurrent:
The explicit representation (15) of $s$ , valid with $ϑ_{1} \in I R$ , shows that $s$ is integrable whenever $ϑ_{1} > \frac{σ^{2}}{2}$ (i.e. $λ_{1} (ϑ) > 1$ ): then $S$ in (14) converges to finite limits $S (\pm \infty)$ as $x \to \pm \infty$ , thus $ξ$ is transient by [GS 72], lemma 3 on p. 117.
In the limiting case $ϑ_{1} = \frac{σ^{2}}{2}$ ( $λ_{1} (ϑ) = 1$ ) to transience, $S$ in (14), of logarithmic growth as $x \to \pm \infty$ , is a bijection onto $I R$ . Thus all cases $ϑ_{1} \leq \frac{σ^{2}}{2}$ ( $λ_{1} (ϑ) \leq 1$ ) lead to recurrence of $ξ$ with invariant measure $\frac{1}{σ^{2}} \frac{1}{s (x)} d x$ , cf. proposition 9.12 a) in [Ho 14]. Ergodicity (positive recurrence) holds if the invariant measure is a finite measure, i.e. for $ϑ_{1} < - \frac{σ^{2}}{2}$ ( $λ_{1} (ϑ) < - 1$ ).

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 $f_{2, ν}$ are bounded functions, thus local $(Q^{ϑ}, I F)$ -martingales defined by (9) are square-integrable martingales with angle bracket (10) under $Q^{ϑ}$ . It remains to prove that $Q^{ϑ}$ -almost surely, the process $J (ϑ)$ takes values in $D^{+}$ , the set of strictly positive definite symmetric matrices in $I R^{(1 + m) \times (1 + m)}$ . For $u \in I R^{1 + m}$ with $| u | = 1$ ,

(18)

u^{⊤} J (ϑ)_{t} u = \frac{1}{σ^{2}} \int_{0}^{t} [u^{⊤} ψ ψ^{⊤} u] (η_{s}) d s = \frac{1}{σ^{2}} \int_{0}^{t} [u^{⊤} ψ]^{2} (η_{s}) d s,, 0 < t < \infty

where we write for short $ψ : I R \to I R^{1 + m}$

for the row vector with entries

f_{1}, f_{2, 1}, \dots, f_{2, m}

. Now

Q^{ϑ}

-almost surely, the range

{η_{s} : 0 \leq s \leq t} \subset I R

of the diffusion path (5) has non-empty interior; on this (random, since defined by the path) open set, the functions

f_{1}, f_{2, 1}, \dots, f_{2, m}

are linearly independent by assumption 1. Hence

Q^{ϑ}

-almost surely, the right hand side in (18) is strictly positive, first pointwise in

u

and

t

, second uniformly on

{| u | = 1} \times [\frac{1}{m}, m]

by continuity, and we let

m \to \infty

□

Proof of proposition 1: Common determinations $Y$ of the stochastic integrals jointly for all $Q_{ϑ}$

do exist since probability measures

Q^{ϑ}

ϑ \in Θ

, are locally equivalent relative to

I F

, see lemma 8.2’ in [Ho 14]. Now we exploit almost sure invertibility of the information

J (ϑ)_{t}

at time

0 < t < \infty

combined with ’inverse bowl shape’ of the log-likelihoods.

□

Proof of proposition 2: By assumption, the starting point $x_{0}$ is an interior point of $A$ . Thus for $0 < t < \infty$ , $A \cap {η_{s} : 0 \leq s \leq t}$ almost surely contains open balls, and (18) remains true with $f \cdot 1_{A}$ in place of $f$ . $□$

Proof for remark 1: ${ˇ ϑ}_{t}$ corresponds to the maximum of the one-dimensional surface

ϑ_{1}^{'} ⟶ (ϑ_{1}^{'} - ϑ_{1}) ˇ S (ϑ)_{t} - \frac{1}{2} (ϑ_{1}^{'} - ϑ_{1})^{2} {ˇ J}_{t}, ϑ_{1}^{'} \in I R .

The representation of estimation errors under $ϑ \in Θ$ follows from (5), (9) and the definition of $Y_{t}$ in proposition 1. By Harris recurrence, the ratio limit theorem holds: for $g, h \in L^{1} (μ^{ϑ})$ with $μ^{ϑ} (h) > 0$ ,

(19)

\frac{\int_{0}^{t} g (η_{s}) d s}{\int_{0}^{t} h (η_{s}) d s} ⟶ \frac{μ^{ϑ} (g)}{μ^{ϑ} (h)} Q^{ϑ} -almost surely as t \to \infty .

This is valid for all $ϑ \in Θ$ and for arbitrary choice of a starting point $x_{0}$ for the process (5). $□$

2 Convergence

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 $ϑ \in Θ$ , we define

(20)

Ψ^{+} (ϑ) := exp {m \sum ν = 1 λ_{2, ν} F_{2, ν} (+ \infty)}, Ψ^{-} (ϑ) := exp {m \sum ν = 1 λ_{2, ν} F_{2, ν} (- \infty)}

(cf. (8), assumption 1 and lemma 1) and introduce a $(0, 1)$ -valued index

(21)

α (ϑ) := \frac{1}{2} (1 - \frac{2 ϑ_{1}}{σ^{2}}) = \frac{1}{2} (1 - λ_{1} (ϑ))

together with weights

(22)

D (ϑ) := \frac{(2 σ^{2})^{1 + α (ϑ)} Γ (α (ϑ))}{2 Γ (1 - α (ϑ))} .

Note that (21) and (22) depend on $ϑ_{1} \in Θ_{1}$ whereas (20) depends on $(ϑ_{2, 1}, \dots, ϑ_{2, m}) \in I R^{m}$ . We shall write $ϕ : I R \to I R^{k}$ for measurable functions whose components $ϕ_{j}$ , $1 \leq j \leq k$ , are such that

(23)

| ϕ_{j} | is locally bounded on I R, and \int_{{| x | > 1}} ϕ_{j}^{2} (x) | x |^{1 - ε} d x < \infty for every ε > 0 .

Functions which satisfy (23) do belong to $L^{2} (μ^{ϑ})$ for every $ϑ \in Θ$ , by lemma 1, and we can define

(24)

Theorem 1: Let $ϕ$ with components $ϕ_{j}$ , $1 \leq j \leq k$ , satisfy (23). With notations (20)–(22) introduce sequences of norming constants by

(25)

α_{n} (ϑ) := n^{α (ϑ)} D (ϑ) / [Ψ^{+} (ϑ) + Ψ^{-} (ϑ)], n \in I N .

Then for every $ϑ \in Θ$ , we have weak convergence in the Skorohod space $D (I R^{k}, I R^{k \times k})$ of

(26)

(\frac{1}{\sqrt{α_{n} (ϑ)}} \int_{0}^{∙ n} ϕ (η_{s}) d m_{s}^{(ϑ)}, \frac{1}{α_{n} (ϑ)} ⟨ \int_{0}^{∙ n} ϕ (η_{s}) d m_{s}^{(ϑ)} ⟩) under Q^{ϑ}

as $n \to \infty$ to

(27)

(Σ^{1 / 2} B \circ V^{α (ϑ)}, Σ V^{α (ϑ)}), Σ = Σ (ϑ, ϕ)

with notation (24). Here $V^{α}$ is a Mittag-Leffler process of index $0 < α < 1$ , the process inverse (i.e. the process of level crossing times) of a stable increasing process $S^{α}$ with index $0 < α < 1$ , and $B$ a $k$ -dimensional Brownian motion which is independent of $V^{α}$ ; thus $B \circ V^{α}$ is Brownian motion subject to independent time change $t \to V_{t}^{α}$ .

The one-dimensional case $k = 1$ of theorem 1 corresponds to theorem A in [HK 03] (note that the invariant measure there has a factor $2$ with respect to our $μ^{ϑ}$ in lemma 1).

Remark 3: The one-sided stable process $S^{α}$ with index $0 < α < 1$ has stationary independent increments with Laplace transform

E (e^{- ζ S_{t}^{α}}) = e^{- t ζ^{α}}, 0 < ζ < \infty .

Its process inverse $V^{α}$ has continuous nondecreasing paths with $V_{0}^{α} = 0$ and ${lim}_{t ↑ \infty} V_{t}^{α} = \infty$ , hence $B \circ V^{α}$ , $V^{α}$ are continuous processes. As a consequence ([JS 87], VI.2.3) of the continuous mapping theorem, theorem 1 implies weak convergence in $I R^{k} \times I R^{k \times k}$ of (26) evaluated at time $t = 1$ to (27) evaluated at time $t = 1$ .

Remark 4: $L (V_{1}^{α})$ is concentrated on $(0, \infty)$ . Whereas $L (V_{1}^{α})$

admits finite moments of arbitrary order

n \in I N

, there is no finite moment of order

- 1

: we have

(28)

\int_{(0, \infty)} L (V_{1}^{α}) (d v) \frac{1}{v} = \infty for 0 < α < 1

which will be of importance below.

2.2 Proof of theorem 1

We formulate a lemma which corresponds to Khasminskii [Kh 80] (see theorem 2.2 in [KY 00] and theorem 1.1 in [Kh 01], or proposition 9.14 in [Ho 14]).

Lemma 3: The path of $ξ$ can be decomposed into iid life cycles $ξ_{]] R_{n}, R_{n + 1}]]}$ , $n \geq 1$ , defined by

R_{n} := inf {t > S_{n} : ξ_{t} < 0}, S_{n} := inf {t > R_{n - 1} : ξ > S^{- 1} (1)}, n \geq 1, R_{0} = 0

(up to an initial segment $ξ_{[[0, R_{1}]]}$ ), with $S^{- 1}$ the function inverse of $S$ in (16) which depends on $ϑ$ . For this decomposition the following holds:

i) for $ϕ$ nonneative and measurable,

E (\int_{R_{n}}^{R_{n + 1}} ϕ (ξ_{s}) d s) = 2 μ^{ϑ} (ϕ) for all n \geq 1;

ii) as $t \to \infty$ , with $α = α (ϑ)$ from (21) and $Ψ^{\pm} = Ψ^{\pm} (ϑ)$ from (20):

P (R_{n + 1} - R_{n} > t) \sim t^{- α} \frac{1}{Γ (α)} {(\frac{1}{2 σ^{2}})}^{α} 2 {Ψ^{+} + Ψ^{-}} .

Proof: 1) The function $S$ in (14) is harmonic for the Markov generator of the diffusion $ξ$ . In a first step we consider $ξ$ transformed by $S$ : $˜ ξ := S \circ ξ$ is a diffusion without drift

d {˜ ξ}_{t} = ˜ σ ({˜ ξ}_{t}) d W_{t}, ˜ σ (˜ x) = σ (s \circ S^{- 1}) (˜ x), ˜ x \in I R

(with constant $σ$ in (5), and $S^{- 1}$ is the function inverse of $S$ ) whose invariant measure is given by

(29)

{˜ μ}^{ϑ} (d ˜ x) = \frac{1}{[˜ σ (˜ x)]^{2}} d ˜ x on (I R, B (I R))

([GS 72], [Kh 80], or proof of proposition 9.12 in [Ho 14]). Calculating from (15) and (16) the function $s \circ S^{- 1}$ and supressing the dependence on $ϑ$ we get

(s \circ S^{- 1}) (˜ x) \sim | ˜ x |^{\frac{- λ_{1}}{1 - λ_{1}}} (1 - λ_{1})^{\frac{- λ_{1}}{1 - λ_{1}}} (Ψ^{\pm})^{\frac{- 1}{1 - λ_{1}}} as ˜ x \to \pm \infty

with notation from (8) and (20), thus with notation (21)

\frac{1}{[˜ σ (˜ x)]^{2}} \sim \frac{1}{σ^{2}} {| ˜ x |}^{λ_{1} / α} (1 - λ_{1})^{λ_{1} / α} (Ψ^{\pm})^{1 / α} as ˜ x \to \pm \infty .

This is the Lebesgue density of the invariant measure ${˜ μ}^{ϑ}$ in (29) for the process $˜ ξ = S \circ ξ$ .

2) We comment on the invariant density for $˜ ξ$ . Multiplying the asymptotic constant by $2$ we define

(30)

A^{\pm} := \frac{2}{σ^{2}} (1 - λ_{1})^{λ_{1} / α} (Ψ^{\pm})^{1 / α}

and have

(31)

\frac{2}{[˜ σ (˜ x)]^{2}} \sim {| ˜ x |}^{λ_{1} / α} A^{\pm} as ˜ x \to \pm \infty .

This is the norming used by Khasminskii for the invariant measure. By (8) and (21), the quotient

β := λ_{1} / α = 2 \frac{λ_{1}}{1 - λ_{1}}

takes values in $(- 1, \infty)$ and is such that

\frac{1}{β + 2} = \frac{\frac{1}{2} (1 - λ_{1})}{λ_{1} + (1 - λ_{1})} = \frac{1}{2} (1 - λ_{1}) = α .

3) Now we apply the results of [Kh 80] quoted in the beginning of section 2.2. The stopping times

R_{n} := inf {t > S_{n} : {˜ ξ}_{t} < 0}, S_{n} := inf {t > R_{n - 1} : ˜ ξ > 1}, n \geq 1, R_{0} = 0

define iid life cycles ${˜ ξ}_{]] R_{n}, R_{n + 1}]]}$ in the path of $˜ ξ$ for $n \geq 1$ , with the following two properties: first,

(32)

E (\int_{R_{n}}^{R_{n + 1}} ˜ ϕ ({˜ ξ}_{s}) d s) = 2 {˜ μ}^{ϑ} (˜ ϕ), n \geq 1

for measurable functions $˜ ϕ : I R \to [0, \infty)$ and ${˜ μ}^{ϑ}$ as in (29); second,

(33)

P (R_{n + 1} - R_{n} > t) \sim t^{- α} \frac{{[α]}^{2 α}}{Γ (1 + α)} {[A^{+}]^{α} + [A^{-}]^{α}} as t \to \infty .

4) The iid life cycles for $˜ ξ = (S (ξ_{t}))_{t \geq 0}$ in step 3) are iid life cycles for $ξ = (ξ_{t})_{t \geq 0}$

R_{n} := inf {t > S_{n} : ξ_{t} < 0}, S_{n} := inf {t > R_{n - 1} : ξ > S^{- 1} (1)}, n \geq 1, R_{0} = 0

for which asymptotics (33) remain unchanged, and change of variables transforms (32) into

E (\int_{R_{n}}^{R_{n + 1}} ϕ (ξ_{s}) d s) = 2 μ^{ϑ} (ϕ) for all n \geq 1 .

Using step 2), the asymptotic constant on the right hand side of (33) equals

\frac{{[α]}^{2 α}}{Γ (1 + α)} {[A^{+}]^{α} + [A^{-}]^{α}} = \frac{1}{Γ (α)} {(\frac{1}{2 σ^{2}})}^{α} 2 {Ψ^{+} + Ψ^{-}}

and lemma 3 is proved by step 3); recall that both $α = α (ϑ)$ and $λ_{1} = λ_{1} (ϑ)$ depend on $ϑ \in Θ$ . $□$

Proof of theorem 1: 1) In a first step, consider convergence of martingales in case $k = 1$ .
Fix $ϑ \in Θ$ and apply theorem 3.1 c) of [HL 03] to one-dimensional measurable functions $ϕ$ which belong to $L^{2} (μ^{ϑ})$ : to the sequence $(R_{n})_{n}$ of renewal times considered in lemma 3 correspond norming sequences

{˜ a}_{n} = \frac{1}{Γ (1 - α) P (R_{2} - R_{1} > n)} = n^{α} \frac{Γ (α)}{Γ (1 - α)} {(2 σ^{2})}^{α} \frac{1}{2 {Ψ^{+} + Ψ^{-}}}

such that

\frac{1}{\sqrt{{˜ α}_{n}}} \int_{0}^{∙ n} ϕ (η_{s}) d m_{s}^{(ϑ)}

converges weakly as $n \to \infty$ in the Skorohod space $D$ of one-dimensional cadlag functions to

\sqrt{˜ c} B \circ V^{α (ϑ)}

where $˜ c = ˜ c (ϑ)$ is given by

(34)

˜ c := E_{ϑ} (\int_{R_{1}}^{R_{2}} ϕ^{2} (η_{s}) d ⟨ m^{(ϑ)} ⟩_{s}) = 2 σ^{2} μ^{ϑ} (ϕ^{2});

here we use (32) together with $d ⟨ m^{(ϑ)} ⟩_{s} = σ^{2} d s$ . Taking into account this factor $2 σ^{2}$ arising on the right hand side of (34) we arrive at the norming sequence $(α_{n})_{n}$ of theorem 1, defined by (25) and (22), and have weak convergence under $Q^{ϑ}$

(35)

\frac{1}{\sqrt{α_{n}}} \int_{0}^{∙ n} ϕ (η_{s}) d m_{s}^{(ϑ)} ⟶ \sqrt{c} B \circ V^{α (ϑ)} (weakly in D as n \to \infty)

with $c := μ^{ϑ} (ϕ^{2})$ as asserted in theorem 1. This proves convergence of martingales in case $k = 1$ .

2) The result proved in step 1) can be extended to case $k > 1$ 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 $E ([S_{1}^{α}]^{p})$ is finite for $0 < p < α$ and infinite for $p = α$ (e.g. [Ho 14], 6.18’). Since $V^{α}$ is the process inverse to $S^{α}$ , the relation

L (V_{1}^{α}) = L ({[\frac{1}{S_{1}^{α}}]}^{α}), 0 < α < 1

holds and proves that $E (\frac{1}{[V_{1}^{α}]})$ equals $\infty$ . $□$

3 LAMN and optimal estimator sequences

For local asymptotic mixed normality see Jeganathan [Je 82], Davies [Da 85], and LeCam and Yang [LY 90]. See also sections 5.1, 6.1 and 7 of [Ho 14].

3.1 Limit distribution for the estimators of section 1.1

Under $ϑ \in Θ$ consider the norming sequence $(α_{n} (ϑ))_{n}$ defined by (25). In notations from (9) and (10), proposition 1 combined with theorem 1 yields convergence in law as $n \to \infty$ of rescaled ML errors

(36)

L (\sqrt{α_{n} (ϑ)} ({ˆ ϑ}_{n} - ϑ) ∣ Q^{ϑ}) ⟶ L (Λ (ϑ)^{- 1 / 2} B (V_{1}^{α (ϑ)}) / V_{1}^{α (ϑ)})

for all $ϑ \in Θ$ , where $B$ is $(1 + m)$ -dimensional Brownian motion independent of the Mittag-Leffler variable $V_{1}^{α (ϑ)}$ , and where $Λ (ϑ)$ is given by

(37)

Λ (ϑ) = \frac{1}{σ^{4}} ⎛ ⎜ ⎜ ⎜ ⎝ \begin{matrix} μ^{ϑ} (f_{1}^{2}) & \dots & μ^{ϑ} (f_{1} f_{2, ν^{'}}) ⋮ & ⋱ & ⋮ μ^{ϑ} (f_{2, ν} f_{1}) & \dots & μ^{ϑ} (f_{2, ν} f_{2, ν^{'}}) \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎠

with $ν$ , $ν^{'}$ ranging from $1$ to $m$ . If we fix as in proposition 2 a compact interval $A$ such that $x_{0} \in i n t (A)$ and replace $f_{1}$ , $f_{2, 1}, \dots, f_{2, m}$ by $f_{1} \cdot 1_{A}$ , $f_{2, 1} \cdot 1$

LAMN in a class of parametric models for null recurrent diffusion

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LAMN in a class of parametric models for null recurrent diffusion

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This week in AI

1 The setting

1.1 Assumptions, null recurrence, likelihoods, some estimators

1.2 The proofs

2 Convergence

2.1 Convergence of martingales together with their angle bracket

2.2 Proof of theorem 1

3 LAMN and optimal estimator sequences

3.1 Limit distribution for the estimators of section 1.1