Local Asymptotic Normality of Infinite-Dimensional Concave Extended Linear Models

TL;DR

This paper establishes local asymptotic normality for M-estimates in infinite-dimensional convex models using Mosco-convergence, enabling analysis despite non-differentiability and constraints.

Contribution

It introduces a novel approach based on Mosco-convergence to prove local asymptotic normality in infinite-dimensional convex models with non-differentiable objectives.

Findings

Proves local asymptotic normality for infinite-dimensional convex M-estimates.

Develops a new technique using Mosco-convergence for non-uniform convergence issues.

Derives asymptotic distribution of likelihood ratio test statistic in Hilbert spaces.

Abstract

We study local asymptotic normality of M-estimates of convex minimization in an infinite dimensional parameter space. The objective function of M-estimates is not necessary differentiable and is possibly subject to convex constraints. In the above circumstance, narrow convergence with respect to uniform convergence fails to hold, because of the strength of it's topology. A new approach we propose to the lack-of-uniform-convergence is based on Mosco-convergence that is weaker topology than uniform convergence. By applying narrow convergence with respect to Mosco topology, we develop an infinite-dimensional version of the convexity argument and provide a proof of a local asymptotic normality. Our new technique also provides a proof of an asymptotic distribution of the likelihood ratio test statistic defined on real separable Hilbert spaces.

Equations

$${\hat{\theta }}_n$$

$$F_n\left(\theta \right)\triangleq \frac{1}{n}\sum _{i=1}^n\rho \left(\theta ,Z_i\right),$$

$$F_0\left(\theta \right)\triangleq {\mathbb{E}}_Z\left[\rho \left(\theta ,Z\right)\right].$$

$$y$$

$${\hat{\theta }}_n$$

$$\underset{\theta \in \Theta }{\sup }\left|\frac{1}{n}\sum _{i=1}^n\rho \left(\theta ,Z_i\right)-{\mathbb{E}}_Z\left[\rho \left(\theta ,Z\right)\right]\right|$$

$$\underset{\theta }{\sup }\left|\frac{1}{n}\sum _{i=1}^n\rho \left(\theta ,Z_i\right)-{\mathbb{E}}_Z\left[\rho \left(\theta ,Z\right)\right]\right|$$

$$n\left[F_n\left({\theta }_0+\frac{1}{\sqrt{n}}t\right)-F_n\left({\theta }_0\right)\right]$$

$$n\left[F_n({\hat{\theta }}_n)-F_n\left({\theta }_0\right)\right]⇝⟨\hat{t},W⟩+\frac{1}{2}⟨V\hat{t},\hat{t}⟩,$$

$$\underset{n\to \infty }{\mathrm{limsup}}\left(\arg \min f_n\right)$$

$$⟨\arg \min f_n,h⟩$$

$$\underset{n\to \infty }{\mathrm{limsup}}{F}_n$$

$$\partial \rho \left(\theta ,Z\right)$$

$${\Psi }_A\left(\theta \right)$$

$$N_A\left(a\right)$$

$${\theta }^{\star }\in \partial {\Psi }_A\left(a\right)$$

$$\iff ⟨\theta -a,{\theta }^{\star }⟩\leqq {\Psi }_A\left(\theta \right)\left(\forall \theta \in A\right)$$

$$\iff ⟨\theta -a,{\theta }^{\star }⟩\leqq 0\left(\forall \theta \in A\right)$$

$$\iff {\theta }^{\star }\in N_A\left(a\right)$$

$$\partial \left(f_1+f_2\right)$$

$$\partial f\left(\theta \right)$$

$$\partial \mathbb{E}\left[\left|y-⟨x,\theta ⟩\right|\right]$$

$$\mathbb{E}\left[\partial \left|y-⟨x,\theta ⟩\right|\right]$$

$$=\mathbb{E}\left[\mathbb{E}\left[x\left\{1-2\mathbb{I}\left(y-⟨x,\theta ⟩\le 0\right)\right\}|x\right]\right]$$

$$=\mathbb{E}\left[x\left\{1-2P_{\epsilon }\left(q-⟨x,\theta ⟩|x\right)\right\}\right].$$

$$J_{\lambda }^{\partial f}\theta$$

$${}^{\forall }\lambda >0,J_{\lambda }^{\partial f}$$

$$\frac{1}{n}\sum _{i=1}^n\partial \rho \left(\theta ,Z_i\right)$$

$$=\partial \mathbb{E}\left[\rho \left(\theta ,Z\right)\right].$$

$$d_G\left(\partial f,\partial g\right)$$

$$H_n\left(\theta ,t\right)$$

$$Q_0\left(t\right)$$

Full text

1 Introduction

We develop an infinite-dimensional version of local asymptotic normality and convexity arguments with non-differentiable objective functions in M-estimation of concave extended linear models. A new approach we propose is based on Mosco convergence that is weaker than uniform convergence in the topological sense. Because of the strength of uniform convergence, it does not fold in infinite dimensional circumstances. In this paper, we give proofs of local asymptotic normality on a real separable Hilbert space.

The basic set-up of the estimation problem we investigate is as follows. Let $\mathscr{H}$ be a real separable Hilbert space with the identical dual $\mathscr{H}^{*}=\mathscr{H}$. We denote the inner product and the associated norm in $\mathscr{H}$ by $\left\langle\cdot,\cdot\right\rangle$ and $\left\|\cdot\right\|$ respectively. Let $\theta$ be a parameter vector in a parameter set $\Theta$ such that $\Theta\subseteq\mathscr{H}$. Suppose we have $n$ observations $Z_{1},\dots,Z_{n}$ that are realizations of a random vector $Z$ on a arbitrary set $E$, and consider an M-estimator of the unknown parameter vector $\theta$ such that

[TABLE]

where $\rho:\Theta\times E\rightarrow(-\infty,\infty]$ is a criterion function. Define the empirical objective function in (1) as

[TABLE]

and its population counterpart as

[TABLE]

We further suppose the minimization problem in (1) is corresponding to a “concave extended linear model”, that is,

$\rho$ is a lower semi-continuous (l.s.c.) convex function with respect to $\theta$ (it is not necessarily smooth, though), 2. 2.

$F_{0}\left(\theta\right)$ is strictly convex in $\theta$ (see, e.g. Huang (2001)) and is uniquely minimized at a (pseudo-) true parameter $\theta_{0}\in\varTheta$.

Compared to the rate of convergence of the the M-estimator for the concave extended linear model, only a few studies have explored its asymptotic distribution and most of them is on the least squares regression case (e.g. Newey (1997), Huang (2003), Belloni et al. (2015)). Recently, Shang and Cheng (2013) obtain a general result on point-wise asymptotic normality of the M-estimator (1) based on functional Bahadur representation. In proving the asymptotic normality, however, they impose the smoothness condition on $\rho$ so that it should be three times continuously differentiable with respect to $\theta$. On the other hand, our new approach does not require the smoothness of $\rho$. The following is our motivating example.

Example** ($L_{1}$ regression.).**

Consider a nonparametric regression model with additive errors:

[TABLE]

where the regressor $x\in\mathscr{H}$ and the error term $\varepsilon$ are mutually independent random variables; $\varepsilon$ is assumed to be homoskedastic; and the conditional median of $\epsilon$ given $x$ is zero, i.e., $\inf\left\{q:\ P_{\varepsilon}\left(q\mid x\right)\geq\frac{1}{2}\right\}=0$ where $P_{\varepsilon}\left(\cdot\mid x\right)$ is the distribution function of $\varepsilon$ conditional on $x$. We are interested in estimating $\theta\in\mathscr{H}$. Suppose we have observations $Z_{i}=\left(y_{i},x_{i}\right):y_{i}\in\mathbb{{R}},x_{i}\in\mathscr{H},i=1,\ldots,n$ independently drawn from the regression model (4). With them, we may estimate $\theta$ via $L_{1}$ minimization with roughness penalty(see, for example Koenker et al. (1994)):

[TABLE]

where $\lambda$ is the smoothing parameter that converges to zero as $n\rightarrow\infty$. Obviously, this example gives the case in which the criterion function $\rho=\left|\cdot\right|$ is not continuously differentiable.

Uniform convergence of the objective function in (1) to its population counterpart:

[TABLE]

guarantees both consistency of of $\hat{\theta}_{n}$ and convergence of the optimal value of the objective function. In order to make the objective function satisfy the uniform convergence, we have to impose some compactness of the parameter space. These assumptions are rather restrictive for fully nonparametric settings. It is because of the theorem by Bakhvalov (Theorem 12.1.1. of Dudley (1999)) . When $\rho=\left|\cdot\right|$ and $\theta$ is in an infinite-dimensional space, we have

[TABLE]

The left-hand side of the inequality does not converge uniformly.

Since $\rho$ is convex, it seems that we may use the convexity lemma (e.g., Pollard (1991) and Kato (2009)) to ensure that point-wise convergence of convex functions implies uniform convergence. In the infinite-dimensional case, however, this argument for uniform convergence may fail. Let $\pi_{n},\ n=1,2,\cdots$ be the sequence of projection operators on $\mathscr{H}$ onto $E_{n}\subset\mathscr{H}$ where $E_{n}\subsetneqq E_{m>n}$. Consider a quadratic form $\left\langle\pi_{n}\theta,\theta\right\rangle$ for $\forall\theta\in\mathscr{H}$ that is considered as a convex function of $\theta$. Then, as $n\rightarrow\infty$, $\left\langle\pi_{n}\theta,\theta\right\rangle$ converges point-wise to $\left\langle\theta,\theta\right\rangle$ but not uniformly.

To solve the aforementioned lack-of-uniform-convergence issue, we shall propose to apply an alternative mode of convergence, Mosco convergence, which is weaker than uniform convergence but still strong enough to enable statistical applications. Mosco convergence of the objective function ensures the convergence of its minimizer (Attouch (1984)). We develop narrow convergence theory with respect to the Mosco metric, see also Geyer (1994), Dupacava and Wets (1988), Molchanov (2005), Knight (2003) and Bucher et al. (2014). There exist alternative forms of convergence that is equivalent to Mosco convergence but more easily verifiable. They include graph convergence (G-convergence) of subdifferential operators and strong convergence of resolvent. We shall explain these key concepts in Section 2. Using these equivalences, we can establish the consistency and narrow convergence of an M-estimator in an infinite-dimensional parameter space. Furthermore, Mosco convergence also ensures the invertibility of the “Hessian” operator.

If the parameter space is weakly compact, Mosco convergence of the convex objective function $F_{n}(\theta)$ in M-estimation ensures that both empirical minimizer $\hat{\theta}_{n}$ and empirical optimal value function $F_{n}(\hat{\theta}_{n})$ will converges to the true parameter $\theta_{0}$ and the true optimal value function $F_{0}(\theta_{0})$ respectively. This property makes it possible to derive the asymptotic distribution of the optimal value function $F_{n}(\hat{\theta}_{n})$. Namely,

The convex objective function $F_{n}\left(\theta\right)$ is locally asymptotically normal at $\theta_{0}$ in Mosco topology:

[TABLE]

where $W$ is a normal random vector in a Hilbert space and $V=\nabla_{\theta}^{2}F_{0}$ is the “Hessian” operator that is almost surely invertible. 2. 2.

The asymptotic distribution of the optimal value function $F_{n}(\hat{\theta}_{n})$ is

[TABLE]

where $\hat{t}=\sqrt{n}(\hat{\theta}_{n}-\theta_{0})$.

As a by-product, the asymptotic distribution of the likelihood ratio statistic can be derived. These results are established in a fully nonparametric setting.

The rest of this paper is organized as follows. In Section 2, we describe the Mosco convergence and introduce the narrow convergence in the Mosco topology. In Section 3, we derive local asymptotic normality of an convex objective function in an infinite-dimensional Hilbert space. We also provide the asymptotic distribution of the likelihood ratio statistic by using the local asymptotic normality. Appendixes give some technical lemmas.

Notations

Let $\rightsquigarrow$ denote narrow convergence and $\xrightarrow{P}$ denote convergence in probability. We use empirical process notation: $\mathbb{G}_{n}\rho=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\rho\left(\theta,Z_{i}\right)-\mathbb{E}\left[\rho\left(\theta,Z_{i}\right)\right]$. We denote $\left\|\theta\right\|$ as $l_{2}$-norm or $L_{2}$-norm of an element of Hilbert space $\theta\in\mathscr{H}$. Let $\theta_{n}\xrightarrow{s}\theta_{0}$ denote convergence in strong topology, e,g,$\left\|\theta_{n}-\theta_{0}\right\|\rightarrow 0$ and $\theta_{n}\xrightarrow{w}\theta_{0}$ denote convergence in weak topology, e,g, $\left\langle\theta_{n},\theta^{*}\right\rangle\rightarrow\left\langle\theta_{0},\theta^{*}\right\rangle$ for all identical dual $\theta^{*}\in\mathscr{H}^{*}\left(=\mathscr{H}\right)$. We denote the limit in weak topology as $\textrm{w-}\lim_{n\rightarrow\infty}\theta_{n}$.

2 Mosco Convergence

First, we introduce a mode of convergence, Mosco convergence, for proper lower semi-continuous (l.s.c.) convex functions on a real separable Hilbert space. For l.s.c. convex functions on a finite dimensional Euclidean space, point-wise convergence is equivalent to locally uniform convergence. For functions defined on an infinite-dimensional space, however, this is not the case. Mosco convergence, on the other hand, still ensures $\arg\min$ convergence of l.s.c. convex functions on an infinite-dimensional space, though it is weaker than locally uniform convergence. In this section, we also provide preliminary results related to Mosco convergence for later use.

Mosco convergence and similar concepts in a non-stochastic environment are considered in Mosco (1969), Attouch (1984) and Beer (1993). Mosco convergence is particularly useful in the context of functional optimization, making it well suited to M-estimation.

Definition 1**.**

[Mosco Convergence]

Let $f_{n}:\mathscr{{H}\rightarrow\left(-\infty,\infty\right],\ }n=1,2,\dots$ be a sequence of proper l.s.c. convex functions. $f_{n}$ is said to be Mosco-convergent to the l.s.c. convex function $f:\mathscr{{H}\rightarrow(-\infty,\infty]}$ if and only if the following two conditions hold.

(M1) For each $\theta\in\mathscr{H}$, there exist a convergent sequence $\theta_{n}\overset{s}{\rightarrow}\theta$ such that ${\displaystyle\limsup_{n}f_{n}\left(\theta_{n}\right)\leq f\left(\theta\right)}$.

(M2) ${\displaystyle\liminf_{n}f_{n}\left(\theta_{n}\right)}\geq f\left(\theta\right)$ whenever $\theta_{n}\overset{w}{\rightarrow}\theta$.

In this paper, we let “$f_{n}\stackrel{{\scriptstyle M}}{{\rightarrow}}f$” denote “$f_{n}$ Mosco-converges to $f$.”

The variational properties of Mosco convergence are given by the following theorem (Theorem 1.10 in Attouch (1984)), which ensures the convergence of both empirical minimizer and empirical minimum value of the objective function to the true ones. Suppose $\arg\min f_{n}\neq\textrm{Ø}$, and existence of $\arg\min f_{n}$ and $\inf f_{n}$ are proved in Appendix A.3.

Theorem 2**.**

We assume the same definitions for $f_{1},f_{2},\cdots$ and $f$. If $f_{n}\stackrel{{\scriptstyle M}}{{\rightarrow}}f$, then

[TABLE]

in the weak topology, e.g.,

[TABLE]

*where the $\limsup$ is defined as

[TABLE]

If there is a weakly compact set $K\subset\mathscr{H}$ such that $\arg\min f_{n}\subset K$ for all $n$ , then $\lim_{n\rightarrow\infty}\left(\inf f_{n}\right)=\inf f.$

It is difficult to prove Mosco convergence directly in general settings. Fortunately, several equivalence conditions for Mosco convergence are known in the literature. One of the most convenient conditions for Mosco convergence is point-wise convergence of subdifferentials of functions.

To deal with this mode of convergence, we introduce several basic tools in convex analysis: subdifferential and resolvent. For more details and proofs on these subjects, see Aubin and Frankowska (1990). For fixed $Z\in E$, we can define a set-valued mapping $\partial\rho\left(\theta,Z\right):\Theta\times E\rightarrow\mathscr{H}$ by

[TABLE]

Such $\partial\rho\left(\theta,\cdot\right)$ is said to be the subdifferential of $\rho$ at $\theta$. For each fixed $\theta$, $\partial\rho\left(\theta,Z\right)$ is considered as a possibly set-valued function of $Z$. We may regard $\partial\rho\left(\theta,Z\right)$ as a generalized derivative of $\rho$ at $\theta$, for each fixed $Z$. If $\rho$ is Gâteaux differentiable at $\theta$ and has a continuous Gâteaux derivative $\nabla\rho\left(\theta\right)$, then $\partial\rho\left(\theta,Z\right)=\nabla\rho\left(\theta,Z\right)$.

Example** ($L_{1}$ regression(continued)).**

The criterion function $\rho\left(\theta,Z\right)=\left|y-\left\langle x,\theta\right\rangle\right|$ is a proper l.s.c. convex function and has the subdifferential such that

[TABLE]

where $\textrm{sgn}\left(y-\left\langle x,\theta\right\rangle\right)=\begin{cases}1,&\text{{if}}\ \left(y-\left\langle x,\theta\right\rangle\right)>0\\ -1,&\text{{if}}\ \left(y-\left\langle x,\theta\right\rangle\right)<0.\end{cases}$

Proof.

Proof is given in Appendix A.1. ∎

Lemma 3** (“Optimization Theory” Indicator Function).**

The indicator function $\Psi_{A}$ is defined by

[TABLE]

where the set $A$ is a convex subset of $\Theta$. The normal cone $N_{A}\left(a\right)$ is defined by

[TABLE]

Then, $N_{A}\left(a\right)=\partial\Psi_{A}\left(a\right)$, where $N_{A}\left(a\right)$ is such that $0\in N_{A}\left(a\right)$.

Proof.

a

[TABLE]

Then, $N_{A}\left(a\right)=\partial\Psi_{A}\left(a\right)$. ∎

Subdifferential operator for proper l.s.c. convex funtions holds distributive law:

[TABLE]

where $f_{1}$ and $f_{2}$ are proper l.s.c. convex functions on $\mathscr{H}$ (see Theorem 3.16. in Phelps (1992)). When $\mathscr{H}$ is real separable, subdifferential operator is exchangeable with respect to integral (Clarke (1983) page 76.):

[TABLE]

Example** ($L_{1}$ regression (continued).).**

The limit criterion $\mathbb{E}\left[\left|y-\left\langle x,\theta\right\rangle\right|\right]$ is convex function and has the subdifferential

[TABLE]

and

[TABLE]

where $P_{\varepsilon}\left(\cdot\mid x\right)$ is the distribution function of $\varepsilon$ conditional on $x$.

In this paper, we assume that the subdifferential $\partial\rho$ is selected and measurable in $Z$. In general, because $\partial\rho$ is a set-valued mapping, the selection is not unique. Nonetheless, we can show that not only such measurable selections exists but also the set of all measurable selector $S_{\partial\rho}$ is identical to $\partial\rho$.

Proposition 4**.**

There exists a measurable selector of the subdifferential $\partial f$, i.e., $S_{\partial f}\neq\emptyset$. Moreover, $S_{\partial f}=\partial f$.

Proof.

Proof is given in Appendix A.2. ∎

Consider a map

[TABLE]

Such a map should be single-valued (on Proposition 3.5.3 in Aubin and Frankowska (1990)). Such $J_{\lambda}^{\partial f},\lambda>0$ are called resolvents of $\partial f$ and denoted by

[TABLE]

The following theorem states the equivalence between Mosco convergence and strong convergence of resolvents and G-convergence of subdifferential operators. The proofs are given in Theorem 3.26. and Theorem 3.66. of Attouch (1984).

Theorem 5**.**

*Let $\mathscr{H}$ be a real separable Hilbert space. Let $\left(f_{n}\right)_{n\in\mathbb{N}}$, $f_{n}:\ \mathscr{H}\rightarrow\left(-\infty,\infty\right],\ ^{\forall}n\in\mathbb{N}$ be a proper l.s.c. convex function. The following statements are equivalent.

(1) $f_{n}\stackrel{{\scriptstyle M}}{{\longrightarrow}}f_{0}$.

(2) ${}^{\forall}\lambda>0$, ${}^{\forall}\theta\in\mathscr{H}$, $J_{\lambda}^{\partial f_{n}}\theta\rightarrow J_{\lambda}^{\partial f}\theta$ strongly in $\mathscr{H}$ as $n$ goes to $\infty$.

(3) $\begin{cases}\partial f_{n}\overset{G}{\rightarrow}\partial f_{0},\\ {}^{\exists}\left(\theta_{0},\eta_{0}\right)\in\partial f_{0}\ ^{\exists}\left(\theta_{n},\eta_{n}\right)\in\partial f_{n}\ such\ that\ \theta_{n}\overset{s}{\rightarrow}\theta_{0},\ \eta_{n}\overset{s}{\rightarrow}\eta_{0},\ f_{n}\left(\theta_{n}\right)\rightarrow f_{0}\left(\theta_{0}\right),\end{cases}$

where $\partial f_{n}\overset{G}{\rightarrow}\partial f_{0}$ means that, for every $\left(\theta_{0},\eta_{0}\right)\in\partial f_{0}$, there exists a sequence $\left(\theta_{n},\eta_{n}\right)\in\partial f_{n}$ such that $\theta_{n}\rightarrow\theta_{0}$ strongly in $\mathscr{H}$, $\eta_{n}\rightarrow\eta_{0}$ strongly in $\mathscr{H}^{*}\left(=\mathscr{H}\right)$.*

Statement (3) in Theorem 5 is called G-convergence of monotone operators. This states that point-wise convergence of all measurable selectors of subdifferential operators is equivalent to Mosco convergence of functionals. When the subdifferential is calculable, point-wise convergence of measurable selectors are easy to verify.

Example** ($L_{1}$ regression(continued)).**

From the foregoing theorems, it will be seen that the law of large numbers(LLN) of subdifferential $\partial\rho\left(\theta\right)$ implies the Mosco convergence. From Lemma 10 and the LLN in Banach spaces for each sequence of mesurable selectors of $\partial\rho\left(\theta\right)$, we have the LLN of subdirrential $\partial\rho\left(\theta\right)$:

[TABLE]

Thus this fact establish the consistency of local functional estimation.

(2) in the above theorem give a metric that induces the Mosco convergence. Based on resolvet, Attouch (1984) (p. 365) gives a metric that induces graph convergence on the space of subdifferential operators:

[TABLE]

for any subdifferential operators $\partial f$ and $\partial g$ where $\lambda_{0}$ is taken strictly positive and $\left\{\theta_{k};k\in\mathbb{N}\right\}$ is a dense subset of $\mathscr{H}$. This metric $d_{G}$ induces the Mosco convergence topology and is complete. Convergence in $d_{G}$ are equivalent to the convergence results in (1)$\sim$(3) in Theorem 5.

Hoffman-Jørgensen weak convergence theory performs in a metric space. Generally, epi-convergence does not usually work with a metric but a semi-metric. Even if functions $f,g$ are different each other, it is possible $f$ epi-converge to $g$ (see, Section 3 in Bucher et al. (2014)). Fortunately in the case where the functional space is constituted by convex functions, we can obtain a metric space as described above. We shall define a weak convergence in the following way.

Definition 6**.**

[Mosco Convergence in Distribution]

A sequence of random elements $f_{n}$ in the space of proper l.s.c. convex functions $\mathscr{H}\rightarrow\left(-\infty,\infty\right]$ is said to be Mosco converges in distribution to the random element $f_{0}$ in the space of proper l.s.c. convex functions if $f_{n}\rightsquigarrow f_{0}$ with metric $d_{G}$. We use the notation $f_{n}\stackrel{{\scriptstyle M}}{{\rightsquigarrow}}f_{0}$.

3 Local Asymptotic Normality

First, we show that the reparametrized objective function admits a certain quadratic expansion. A common starting point in developing an asymptotic distribution theory for an M-estimator is to define a centered stochastic process based on the objective function. Recall that $F_{n}(\theta)=\frac{1}{n}\sum_{i}\rho\left(\theta,Z_{i}\right)$ is the objective function for the M-estimator (1). We may define such a centered stochastic process as

[TABLE]

[TABLE]

where $W$ is an $N\left(\boldsymbol{0},A\right)$ random vector in a Hilbert space and $V$ is a “Hessian” operator. $H_{n}\left(\theta_{0},t\right)$ is interpreted as the log likelihood ratio for hypothesis testing against the local alternative, i.e., $\mathcal{{H}}_{0}:\ \theta=\theta_{0};\ \mathcal{{H}}_{1}:\ \theta=\theta_{0}+\frac{1}{\sqrt{{n}}}t$. Define the locally asymptotically quadratic (LAQ) as follows.

Definition 7** **(LAQ LeCum and

Yang (2000)(p. 120))).

The convex objective function $F_{n}\left(\theta\right)$ is said to be locally asymptotically quadratic at $\theta$ if there exists a random matrix $V_{n,\theta}$ and a random vector $\Delta_{n,\theta}$ such that

[TABLE]

and the matrix $V_{n,\theta}$ and their limit $\left(V_{n,\theta}\rightsquigarrow\right)V_{\theta}$ are almost surely invertible.

Remark*.*

Recall that locally asymptotically mixed normality (LAMN) is equivalent to LAQ with a restriction: $\Delta_{n,\theta},V_{n,\theta}$ converge to normal distributions. Locally asymptotically normality (LAN) is equivalent to LAMN with the limiting matrix $V_{\theta}$ is deterministic.

3.1 Second Order Differentiability

In typical situations, we assume that the function $F_{0}$ has a quadratic expansion at $\theta_{0}$ and their Hessian is often supposed to be continuously invertible (Theorem 3.3.1. of van der Vaart and Wellner (1996)). In an infinite-dimensional case, the assumption that the Hessian operator is continuously invertible is harder to ascertain. However, if the convex function $F_{0}$ has a* *generalized second order differentiability (defined later), its “generalized Hessian” is continuously invertible.

Define the Young-Fenchel conjugate $f^{*}$ of convex function $f$ as

[TABLE]

The conjugate $f^{*}$ has a strong link between a convex function $f$ in the second order differentiability. Recall the case of a convex function defined on finite dimensional parameters. A convex function $f$ defined on the Euclid space $\mathbb{R}^{d}$ is second order differentiable and the Hessian $\nabla^{2}f\left(\theta\right)$ of $f$ at $\theta$ is nondegererate. Then the conjugate function $f^{*}$ is second order differentiable at $y=\nabla f\left(\theta\right)$, and its Hessian $\nabla^{2}f^{*}\left(\eta\right)$ at $y$ is the inverse of $\nabla^{2}f\left(\theta\right)$, i.e.,

[TABLE]

In order to maintain a duality-type of this relation in an infinite-dimensional space, we shall define the second order differential concepts based on Mosco convergence. Mosco convergence ensures the continuity of this type of conjugation (Kato (1989) and Borwein and Noll (1994)).

Define second difference quotient of $f$ at $\theta\in\mathscr{H}$ relative to $\eta^{*}\in\partial f\left(\theta\right)$ as

[TABLE]

and define a purely quadratic continuous convex function as

[TABLE]

where $V$ is a closed symmetric positive linear operator. $f$ is said to have generalized second order differentiability* ***at $\theta$ relative to $\eta^{\star}\in\partial f\left(\theta\right)$ if there exists a purely quadratic function $q$ such that the second order difference quotient $\Delta_{f,\theta,\eta,t}\left(\cdot\right)$ converges to $q\left(\cdot\right)$ in the Mosco sense, i.e.,

[TABLE]

The closed symmetric positive linear operator $V$ is called the generalized Hessian of $f$ at $\theta$ relative to $\eta\in\partial f\left(\theta\right)$.

Mosco convergence is invariant under Young-Fenchel conjugation, so that Mosco convergence of $\Delta_{f,\theta,\eta,t}\left(h\right)$ is equivalent to Mosco convergence of $\left(\Delta_{f,\theta,\eta,t}\left(h\right)\right)^{*}=\Delta_{f*,\eta,\theta,t}\left(h\right)$. And generalized Hessian of $f^{*}$ at $\eta$ relative to $\theta\in\partial f^{*}\left(\eta\right)$ is $V^{-1}$.

Next, we derive sufficient conditions under which the objective function of M-estimation has generalized second order differentiability. $\partial f$ is called** weak Gâteaux differentiable at $\theta$* **if there exists a bounded linear operator $T:\mathscr{H}\rightarrow\mathscr{H}^{*}$ such that

[TABLE]

in the weak* sense for any fixed $h\in\mathscr{H}$ and all $\eta_{t}^{*}\in\partial f\left(\theta+th\right)$, $\eta^{*}\in\partial f\left(\theta\right)$ where $\partial f\left(\theta\right)$ must consist of a single element $\eta^{*}$ . We use the notation $T=\nabla\partial f\left(\theta\right)$ for the operator $T$. For the generalized differentiability, we quote the following result of Borwein and Noll (1994).

Theorem 8**.**

*(a variant of Propotion 6.4. of Borwein and Noll (1994))

Let $\left(Z,\mathcal{Z},\mathbb{P}_{Z}\right)$ be a probability space and $\Theta\subseteq\mathscr{H}$ be a separable Hilbert space. Suppose $\rho:\Theta\times Z\rightarrow\left(-\infty,\infty\right]$ is measurable on $\left(Z,\mathcal{{Z}},\mathbb{{P}}_{Z}\right)$ and convex at any $\theta\in\Theta$ and define a closed convex integral functional $f$ on $\Theta\subset\mathscr{H}$ as*

[TABLE]

Then $f$ is generalized second order differentiable at $\theta$ if and only if $\partial\rho$ is weak Gâteaux differentiable and*

[TABLE]

Example** ($L_{1}$ regression(continued)).**

Let $Z=\left(Y,X\right)$ be a random vector, where $Y$ is real-valued while $X$ is the covariate and $X\in\mathscr{H}$. Note that objective function of $L_{1}$ regression is

[TABLE]

Then, $L_{1}$ regression objective function $F\left(\theta\right)$ is generalized second order differentialbe at $\theta$ if and only if $\partial\mathbb{E}\left[\left|Y-\left\langle x,\theta\right\rangle\right|\left|X\right.\right]$ is weak* Ĝateaux differentiable and

[TABLE]

From (6), weak* Gâteaux differentiability of $\partial\mathbb{E}\left[\left|Y-\left\langle x,\theta\right\rangle\right|\left|X\right.\right]$ at $\theta$ is equivalent to the Gâteaux differentiability of the distribution function $F_{e}\left(q-\left\langle x,\theta\right\rangle\left|x\right.\right)$ at $\theta$. If the distribution function $F_{e}\left(q-\left\langle x,\theta\right\rangle\left|x\right.\right)$ is Gâteaux differentiable at $\theta$, essential boundedness of $\textrm{ess}\sup_{x\in X}\left|\nabla\partial\mathbb{E}\left[\left|Y-\left\langle x,\theta\right\rangle\right|\left|X\right.\right]\right|<\infty$ will be automatically satisfied.

Therefore, in order to obtain invertiblity of “generalized Hessian”, we impose the following assumption on $\rho$:

Assumption**.**

*A

$\partial\rho\left(\cdot\right)$ is weak* Gâteaux differentiable at $\theta_{0}$ and*

[TABLE]

This assumption is a “low-level” condition which are sufficient for locally asymptotically quadratic at $\theta_{0}$ than that of Geyer (1994). Of course, this result is attributed to the convexity of the objective function.

3.2 LAN

Define auxiliary stochastic process as

[TABLE]

We also impose the following assumption. Considering Proposition 4 : the set of all mesurable selectors of a subdifferential coincides with its own subdifferential, we denote any measurable selector of $\partial\rho\left(\cdot\right)$ as itself.

Assumption**.**

*B

Every mesurable selector in $\partial\rho\left(\theta,Z\right)$ has a bounded variance: $\forall\theta\in\Theta$, $\mathbb{E}\left[\left\|\partial\rho\left(\theta,Z\right)\right\|^{2}\right]<\infty$, and there is a sequence of mesurable selectors satisfying a central limit theorem in the Hilbert space:*

[TABLE]

for some trace class covariance operator $A$.

Proposition 9**.**

LAN

$H_{n}\left(t\right)$ Mosco-converges to $G_{n}^{\prime}\left(t\right)$ in probability. 2. 2.

$G_{n}^{\prime}\left(t\right)$ converges in law to $Q_{0}\left(t\right)$. Then, $H_{n}\left(t\right)$ Mosco-converge in law to $Q_{0}\left(t\right)$.

Proof.

We shall prove the first statement. In order that $H_{n}\left(t\right)$ converges in Mosco to $G_{n}\left(t\right)$, we will apply Theorem 5 to $H_{n}\left(t\right)$ and $G_{n}\left(t\right).$ All we have to do is to show the graph convergence of the subdifferential $\partial H_{n}\left(t\right)$ to $\partial G_{n}\left(t\right)$ in probability. Considering proposition 4, we denote any measurable selector of $\partial\rho\left(\cdot\right)$ as itself in the following proof below. Calculate subdifferential of $H_{n},G_{n}$ with respect to $t$, we obtain

[TABLE]

Recall $\partial f_{n}\overset{G}{\rightarrow}\partial f_{0}$ means that for every $\left(\theta_{0},\eta_{0}\right)\in\partial f_{0}$, there exists a sequence $\left(\theta_{n},\eta_{n}\right)\in\partial f_{n}$ such that $\theta_{n}\rightarrow\theta_{0}$ strongly in $\mathscr{H}$, $\eta_{n}\rightarrow\eta_{0}$ strongly in $\mathscr{H}^{*}\left(=\mathscr{H}\right)$. $\partial H_{n}\overset{G}{\rightarrow}\partial G_{n}$ means that there exists a sequence of measurable selectors of $\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\partial\rho\left(\theta_{0}+\frac{1}{\sqrt{n}}t,Z_{i}\right)$ such that

[TABLE]

strongly in $\mathscr{H}$.

The random variable

[TABLE]

converges monotonically to non-negative random variable. Because $F_{0}\left(\theta\right)=\mathbb{E}\left[\rho\left(\theta\right)\right]$ is second order differentiable in the generalized sense,

[TABLE]

so,

[TABLE]

Fix $t$ and define a (selected) random variable $\xi_{ni}$ by

[TABLE]

Note that

[TABLE]

where $\mathbb{E}\left[\frac{1}{\sqrt{n}}\partial\rho\left(\theta_{0}+\frac{1}{\sqrt{n}}t,Z\right)\right]$ is singleton. Therefore, for any selected $\xi_{ni}$,

[TABLE]

and

[TABLE]

Since $\xi_{n1},\dots,\xi_{nn}$ are i.i.d., we have

[TABLE]

for any selected $\xi_{ni}$. We have the equality

[TABLE]

By weak* differentiability of $\mathbb{E}\left[\partial\rho\right]$ at $\theta_{0}$, the limit of any measurable selector of $\partial\rho\left(\theta_{0}+\frac{1}{\sqrt{n}}t,Z_{i}\right)-\partial\rho\left(\theta_{0},Z_{i}\right)$ has expectation zero. From the Assumption B : for every measurable selector $\mathbb{E}\left[\left\{\partial\rho\left(\theta,Z_{i}\right)\right\}^{2}\right]<\infty$ for each $\theta$ in the neighborhood of $\theta_{0}$ and from Lebesgue dominated convergence theorem, we have

[TABLE]

Thus, ${\displaystyle\textrm{Var}\left[\sum_{i=1}^{n}\xi_{ni}\right]\leq\sum_{i=1}^{n}\mathbb{E}\left[\xi_{ni}^{2}\right]\rightarrow 0}$. By Chebyshev inequality, we have

[TABLE]

for fixed $t$. Then, $H_{n}\left(t\right)$ converges in Mosco to $G_{n}\left(t\right)$ in probability.

From Assumption A, $F_{0}$ is second order differentiable in generalized sense:

[TABLE]

Therefore, combining aforementioned result, we obtain the result that $H_{n}\left(t\right)$ Mosco-converges to $G_{n}^{\prime}\left(t\right)$ in probability.

The second statement of Proposition 9 is derived from Assumption B and a.s. representation theorem (Theorem 1.10.4. of van der Vaart and Wellner (1996)). We get

[TABLE]

and an almost sure representation $\tilde{\xi}_{n}\rightarrow\tilde{\xi}\ a.s.$, where $\tilde{\xi}_{n}$ has the same law as $\xi_{n}$ and $\tilde{\xi}$ the same law as $\xi$. This provide the Mosco convergence in distribution of $G_{n}^{\prime}$ to $Q_{0}$. ∎

The aforementioned proposition achieves mosco convergence of $H_{n}$ to its limit $Q_{0}$. Note that $t=\sqrt{n}\left(\theta-\theta_{0}\right)$ minimizes $H_{n}\left(t\right)$.

Next, we will also show convergence of the minimizer of $H_{n}$ to that of $Q_{0}$, provided that the minimizer is almost surely unique. This follows from the following lemma.

Lemma 10**.**

The minimizer of the function $Q_{0}\left(t\right)=\left\langle t,W\right\rangle+\frac{1}{2}\left\langle Vt,t\right\rangle$ is single valued.

Proof.

Let $t_{0}=\arg\min_{t}Q_{0}\left(t\right)$. Suppose there exists $t_{1}\left(\neq t_{0}\right)$ such that

[TABLE]

Then,

[TABLE]

This means $Q_{0}\left(\frac{t_{1}+t_{0}}{2}\right)<\alpha$, which is contradiction. ∎

We apply the previous results to consider the asymptotic distribution of $\sqrt{n}\left\langle\hat{\theta}-\theta_{0},\theta^{*}\right\rangle$ in the weak topology.

Corollary 11**.**

*Asymptotic Normality

Let $W$ be an $N\left(0,A\right)$ distribution. Under Assumption A and B, we obtain the asymptotic distribution of $\sqrt{n}\left\langle\hat{\theta}_{n}-\theta_{0},\theta^{*}\right\rangle$ as following;*

[TABLE]

where $V^{-1}$ is generalized Hessian of Young-Fenchel conjugate of $F_{0}\left(\theta\right)$.

Proof.

From Proposition 9, $H_{n}\left(\theta_{0},\hat{t}\right)$ converges weakly to $Q_{0}\left(t\right)$ in Mosco topology. Applying a.s. representation theorem(Theorem1.10.4 in van der Vaart and Wellner (1996)) we get an almost sure representation $H_{n}\stackrel{{\scriptstyle M}}{{\longrightarrow}}Q_{0}\ a.s.$. By Theorem 5 we have

[TABLE]

in the weak topology. This provide

[TABLE]

∎

Example** ($L_{1}$ regression (continued).).**

Suppose the distribution function $F_{e}\left(q-\left\langle x,\theta\right\rangle\left|x\right.\right)$ is Gâteaux differentiable at $\theta$ and denote their differential as operator $V$. Under Assumption A and B, for any $x_{0}\in\mathscr{H}$,

[TABLE]

For the implement, we need a consistent estimators of the generalized Hessian. From the fact of the properties of the generalized differential, the natural candidates are

[TABLE]

in the weak* sense for any fixed $h\in\mathscr{H}$ and all $\hat{\eta}_{k_{n}}^{\star}\in\partial f\left(\hat{\theta}+k_{n}h\right)$, $\hat{\eta}^{\star}\in\partial f\left(\hat{\theta}\right)$ .

3.3 Likelihood Ratio Test Statistic

Using the previous LAN result, we derives the asymptotic distribution of the likelihood ratio statistic. Let $A_{n}=\sqrt{n}\left(\Theta-\theta_{0}\right)$ and $A_{n,0}=\sqrt{n}\left(\Theta_{0}-\theta_{0}\right)$. The likelihood ratio statistic is written by the form

[TABLE]

By the previous LAN result, for large $n$, the likelihood ratio process is similar to the same as in the normal experiment. And by the Mosco convergence argument in theorem 5, if the parameter space is weakly compact, the empirical optimal value of convex function achieve the true optimal.

Assumption**.**

*C

The parameter set $\Theta$ is weakly compact. In a Hilbert space setting $\Theta\subset\mathscr{H}$, weakly compactness is equal to boundedness: for all $\theta\in\Theta$,there exists constant $C$ such that $\left\|\theta\right\|\leq C$.*

Lemma 12**.**

Let $W$ be an $N\left(0,A\right)$ distribution and repeat (7);

[TABLE]

Let $\hat{t}=\sqrt{n}\left(\hat{\theta}_{n}-\theta_{0}\right)$ denote this minimizer. Under Assumption A-C, the asymptotic distribution of the optimal value function

[TABLE]

is the distribution of $Q_{N}\left(\hat{t}\right)$.

Proof.

From Proposition 9, $H_{n}\left(\theta_{0},\hat{t}\right)$ converges weakly to $Q_{N}\left(t\right)$ in Mosco topology. Applying a.s. representation theorem(Theorem1.10.4 in van der Vaart and Wellner (1996)) we get an almost sure representation $H_{n}\stackrel{{\scriptstyle Mosco}}{{\longrightarrow}}Q_{N}\ a.s.$. By Theorem 2 and Assumption C, we have

[TABLE]

This provide the optimal value of function $H_{n}$ converges weakly to $Q_{N}$. ∎

Define an objective function with convex constraint $G\left(\theta\right)$ from $\mathscr{H}$ to $\left(-\infty,\infty\right]$ by

[TABLE]

where $\Psi_{A}$ is defined by

[TABLE]

and $A$ is convex. Because $F_{n}$ and $\Psi_{A}$ are convex function, $G_{n}\left(\theta\right)$ are also convex function with respect to $\theta$ for all $n$. Redefine (7), (8) as

[TABLE]

where $T_{A}\left(\theta\right)$ is tangent cone:

[TABLE]

From the result of lemma (3) and lemma (12), we obtain the asymptotic distribution of the optimal value function

[TABLE]

The above result yeilds the asymptotic distribution of the likelihood ratio statistics $\Lambda_{n}$. The proof strategy is based on van der Vaart (1998), Chapter 16, Theorem 16.7.

Proposition 13**.**

Assume the parameter spaces $\Theta$ and $\Theta_{0}$ is convex. And assume Assumption A-C. If the sets $A_{n}$ and $A_{n,0}$ converge to sets $A$ and $A_{0}$, then the sequence of likelihood ratio statistics $\Lambda_{n}$ converges under $\theta_{0}+\frac{t}{\sqrt{n}}$ in distribution to

[TABLE]

where $W$ is an $N\left(\boldsymbol{0},A\right)$ random vector.

Proof.

By Lemma 12 and simple algebra

[TABLE]

the proposition follows by the continuous mapping theorem. ∎

Example** ($L_{1}$ regression(continued)).**

Consider a likelihood ratio statistics for testing the value of $\left\langle\theta_{0},x_{0}\right\rangle$ at any $x_{0}\in E$. For some prespecified point $\left(x_{0},c\right)$, we consider the following hypothesis:

[TABLE]

The objective function under the null constrained is defined as

[TABLE]

where $\theta^{H_{0}}\in H_{0}=\left\{\theta\in\Theta:\left\langle\theta_{0},x_{0}\right\rangle\leq 0\right\}$. Note that the set $H_{0}$ is convex. We define the generalized likelihood ratio test statistic as

[TABLE]

where $\hat{\theta}^{H_{0}}$ is the M-estimator under convex constraint:

[TABLE]

If the null the interior of the hypothesis $H_{0}$ contains the true parameter $\theta_{0}$, the sequence of $\Lambda_{n}$ converges to zero in distribution. This means that an error of the first kind converges to zero under that the null hypothesis is true. If the true parameter $\theta_{0}$ belongs to the boundary: $\left\langle\theta_{0},x_{0}\right\rangle=0$, the sets $\sqrt{n}\left(\Theta_{0}-\theta_{0}\right)$ converge to the $H_{0}=\left\{\theta:\ \left\langle\theta,x_{0}\right\rangle\leq 0\right\}$. The sequence of $\Lambda_{n}$ converges in distribution to the distribution of the square distance of a standard normal vector to the half-space $V^{\frac{1}{2}}H_{0}=\left\{\theta:\ \left\langle\theta,V^{-\frac{1}{2}}x_{0}\right\rangle\leq 0\right\}$, that is the distribution of $\left(W\lor 0\right)^{2}$.

Appendix A Appendix

A.1 Proof of Subdifferential Calculus of $\rho=\left|y-\left\langle x,\theta\right\rangle\right|$

Here we show the subdifferential calculus of $\rho=\left|y-\left\langle x,\theta\right\rangle\right|$. We use the following lemma.

Lemma 14**.**

The subdifferential of $\left\|\theta\right\|=\left\langle\theta,\theta\right\rangle$ is $\partial\left\|\theta\right\|=\left\{\theta\right\},\ \theta\in\mathscr{H}$ .

Proof.

For $\theta\in\mathscr{H}$,

[TABLE]

then $\partial\left\|\theta\right\|=\left\{\theta\right\}$. ∎

Proposition** (Subdifferential Calculus of $\rho=\left|y-\left\langle x,\theta\right\rangle\right|$).**

The criterion function $\rho\left(\theta,Z\right)=\left|y-\left\langle x,\theta\right\rangle\right|$ is a proper l.s.c. convex function and has the subdifferential such that

[TABLE]

Proof.

Let $t\in\left[-1,1\right]$, $\theta=tx$. For all $\zeta\in\mathscr{H}$,

[TABLE]

Then, $\theta=tx\in\partial\rho\left(y-\left\langle x,\theta\right\rangle=0\right)$ and $\left[-1,1\right]x\subset\partial\rho\left(y-\left\langle x,\theta\right\rangle=0\right)$.

Next, we shall show the inverse inclusion: $\partial\rho\left(y-\left\langle x,\theta\right\rangle=0\right)\subset\left[-1,1\right]x$. Let $\theta\in\partial\rho\left(y-\left\langle x,\theta\right\rangle=0\right)$ and assume $\theta\neq x$. From $\theta\in\partial\rho\left(y-\left\langle x,\theta\right\rangle=0\right)$, we have

[TABLE]

From now on, set $H=\left\{\eta\in\mathscr{H}:\ \left\langle x,\eta\right\rangle=y\right\}$ and $G=\left\{\eta\in\mathscr{H}:\ \left\langle\eta,\theta\right\rangle=\left\langle\theta,\theta\right\rangle\right\}$, we shall show that $H=G$. When $\dim\left(\mathscr{X}\right)=1$, $H=G=\left\{\frac{y}{x^{*}}\right\}$. Assume $\dim\left\{\mathscr{H}\right\}>2$. First $\eta\in H\Rightarrow\eta\in G$, pick $\eta\in H$: $\left\langle x,\eta\right\rangle=y$ we have $\eta=\theta$, so $\left\langle\eta,\theta\right\rangle=\left\langle\theta,\theta\right\rangle$. Then, $H\subset G$. We shall show the inverse inclusion $G\subset H$. Assume $\eta\in G$ and $\eta\notin H$. Because $\theta\neq x$, there exists $u\in\mathscr{H}$ such that $\left\langle\theta,u\right\rangle\neq y$. Put $p=\left\langle x,\eta\right\rangle u-\left\langle x,u\right\rangle\eta+\theta$, because $u$ and $\eta$ are linear independent, $p\neq\theta$. On the other hand

[TABLE]

This is contradiction, therefore $G\subset H$. Finally, we have $G=H$.

Now, set

[TABLE]

Then, we have

[TABLE]

Furthermore $x^{\prime}\in H\Rightarrow x^{\prime}\in G$. Therefore,

[TABLE]

and we get $\left\langle\theta,\zeta-\theta\right\rangle=t\left\langle x,\zeta-\theta\right\rangle$ where $t=\frac{\left\langle\theta,\ v-\theta\right\rangle}{y-\left\langle x,v\right\rangle}\neq 0$. Because of $\eqref{(*)}$, $\left\langle\theta,\ v-\theta\right\rangle\leq\left|y-\left\langle x,v\right\rangle\right|$ and

[TABLE]

Since $\left\langle\theta,\zeta-\theta\right\rangle\neq 0$,$\left\langle x,\zeta-\theta\right\rangle\neq 0$. We have $\left|\left\langle\theta,\ v-\theta\right\rangle\right|\leq\left|y-\left\langle x,v\right\rangle\right|$, $\left|t\right|\leq 1$. Therefor, $\partial\rho\left(y-\left\langle x,\theta\right\rangle=0\right)\subset\left[-1,1\right]x$. ∎

A.2 Proof of Proposition 4

Set the following notation;

$\left(\Omega,\mathcal{F},\mathbb{P}\right)$: probability triple

$\left(\mathscr{H},\mathcal{H}\right)$: real separable Hilbert space with Borel $\sigma$-field

$2^{\mathscr{H}}$: the family of all nonempty subsets of $\mathscr{H}$

$F:\Omega\rightarrow 2^{\mathscr{H}}$: set-valued function.

The inverse image $F^{-1}\left(X\right)$ is defined by

[TABLE]

A set-valued function $F:\Omega\rightarrow 2^{\mathscr{X}}$ is called measurable if $F^{-1}\left(X\right)$ is measurable for every closed subset $X$ of $\mathscr{X}$. For $1\leq p\leq\infty$ define a selection of $F$ by

[TABLE]

The key notion of set-valued mesurable mapping is decomposability.

Definition 15**.**

Decomposability [Section3 in Hiai and Umegaki (1977)]

Let $M$ be a set of measurable functions $f:\Omega\mapsto\mathscr{H}$. $M$ is called decomposable with respect to $\mathcal{F}$ if $f_{1},f_{2}\in M$ and $A\in\mathcal{F}$ implies

[TABLE]

For proof of Proposition 4, we need lemmas from Hiai and Umegaki (1977).

Lemma 16**.**

[Lemma1.1. in Hiai and Umegaki (1977)}

Let $F$ be measurable set-valued function. If $S_{F}^{p}$ is nonempty, then there exists a sequence $\left\{f_{n}\right\}\in S_{F}^{p}$ such that $F\left(\omega\right)=\textrm{cl}\left\{f_{n}\left(\omega\right)\right\}$ for all $\omega\in\Omega$.

Lemma 17**.**

*[Lemma 2.1. in Hiai and Umegaki (1977)]

Let $\phi:\Omega\times\mathscr{H}$ be $\mathcal{F}\otimes\mathcal{H}$-measurable. Assume $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ is complete and $\phi\left(\omega,\theta\right)$ is l.s.c. in $\theta$ for every fixed $\omega$. Then the function*

[TABLE]

is measurable.

Lemma 18**.**

[Theorem 3.1. in Hiai and Umegaki (1977)}

$M=S_{F}$* if and only if $M$ is decomposable.*

For the set-valued random variables the following Theorem and definition were given by Hiai and Umegaki

Proposition**.**

4* There is a measurable selector of subdifferential $\partial f$ i.e., $S_{\partial f}\neq\emptyset$. And the set of all measurable selector is identical to subdifferential $\partial f$: $S_{\partial f}=\partial f$.*

Proof.

Let $h\left(\gamma,z\right)$ as

[TABLE]

Fix $\alpha$. $\gamma$ is a subdifferential of $f\left(\cdot,z\right)$ at $\alpha$ iff $h\left(\gamma,z\right)\geq 0$. For every $z$, $h\left(\gamma,\cdot\right)$ is measurable. From Lemma 17 $\gamma\left(\cdot\right)$ is measurable.

Let $\gamma_{1}\left(\cdot\right)$ and $\gamma_{2}\left(\cdot\right)$ be measurable selector of subdifferential $\partial f\left(\alpha,\cdot\right)$ satisfying

[TABLE]

From the following inequality

[TABLE]

$\partial f\left(\alpha,\cdot\right)$ is decomposable. Therefore, from Lemma 18 and Lemma 16, $S_{\partial f}=\partial f$. ∎

A.3 A Existence of Minimum

Proposition**.**

*Existence of Minimum

Suppose $f:\Theta\rightarrow(-\infty,\infty]$ is a lower semi-continuous convex (l.s.c.) functional and its domain $\Theta$ is bounded. Then there exists $\arg\min_{\theta}f\left(\omega,\theta\right)$ and $\inf_{\theta}f\left(\omega,\theta\right)$.*

Proof.

Let $C$ be a convex subset of a Banach space. From the separation theorem, $C$ is closed in norm topology if and only if $C$ is closed in the weak topology(Correspondence of closedness). $f$ is lsc on $\Theta$ in the norm topology if and only if $f$ is lsc in the weak topology.

For each $a\in\mathbb{R}$ put

[TABLE]

$G_{a}$ is open in the weak topology and $\Theta=\bigcup_{a\in\mathbb{R}}G_{a}$. Since $\Theta$ is weakly compact, there is finite subcover such that

[TABLE]

Putting $a_{0}=\min\left\{a_{1},\cdots,a_{n}\right\}$, we have $f\left(\theta\right)>a_{0}$ for all $\theta\in\Theta$. There exists a real number $b=\inf\left\{f\left(\theta\right):\ \theta\in\Theta\right\}$.

Suppose $f\left(\theta\right)>b$ for all $\theta\in\Theta$, then

[TABLE]

Since $\Theta$ is weakly compact,

[TABLE]

Put $b_{0}=\min\left\{b+\frac{1}{n_{1}},\cdots,b+\frac{1}{n_{m}}\right\}$, we have $f\left(\theta\right)>b_{0}$ for all $\theta$. Therefore we have

[TABLE]

This is a contradiction. ∎

Acknowledgements

This research is supposed by grant-in-aid for JSPS Fellows (DC1, 20137989).