# A Review of Brown 1971 (in)admissibility results under scale mixtures of
Gaussian priors

Yuzo<EMAIL_ADDRESS>[ William, E.
<EMAIL_ADDRESS>[ Kobe University and Rutgers
University
Kobe University and Rutgers University

###### Abstract

Brown’s 1971 paper “Admissible estimators, recurrent diffusions and insoluble
boundary value problems” is a landmark in the admissibility literature. It
nearly completely settles the issue of admissibility/inadmissibility for
estimating the mean of a multivariate normal distribution with identity
covariance under sum of squared error loss. We revisit this wonderful tour de
force on its 50th anniversary and present an alternative and more direct proof
of the result for generalized Bayes estimators corresponding to priors which
are a subclass of scale mixtures of spherical normals.

62C15,

62J07,

admissibility,

Bayes,

###### keywords:

[class=MSC]

###### keywords:

and

##  1 Introduction

Brown’s 1971 paper “Admissible estimators, recurrent diffusions and insoluble
boundary value problems” is a landmark in the admissibility literature. It
nearly completely settles the issue of admissibility for estimating the mean
of a multivariate normal distribution with identity covariance under sum of
squared error loss. We revisit this wonderful tour de force on its 50th
anniversary and present an alternative and more direct proof of the result for
generalized Bayes estimators corresponding to priors which are a subclass of
scale mixtures of spherical normals.

Specifically, let

$\displaystyle X\sim N_{d}(\mu,I)$

with density

$\displaystyle\phi(x-\mu)=\frac{1}{(2\pi)^{d/2}}\exp\left(-\frac{\|x-\mu\|^{2}}{2}\right).$

Consider estimation of $\mu$ under quadratic loss $\|\delta-\mu\|^{2}$. Let
$\Pi(\mathrm{d}\mu)$ and $m_{\Pi}$ be the prior measure and the corresponding
marginal density given by

$\displaystyle m_{\Pi}(x)=\int\phi(x-\mu)\Pi(\mathrm{d}\mu).$ (1.1)

Then the (generalized) Bayes estimator under $\Pi(\mathrm{d}\mu)$ is given by

$\displaystyle\delta_{\Pi}$
$\displaystyle=\frac{\int\mu\phi(x-\mu)\Pi(\mathrm{d}\mu)}{\int\phi(x-\mu)\Pi(\mathrm{d}\mu)}$
(1.2)
$\displaystyle=x+\frac{\int(\mu-x)\phi(x-\mu)\Pi(\mathrm{d}\mu)}{\int\phi(x-\mu)\Pi(\mathrm{d}\mu)}$
(1.3) $\displaystyle=x+\nabla_{x}\log m_{\Pi}(x).$ (1.4)

We are interested in determining admissibility/inadmissibility of generalized
Bayes estimators, for which Brown (1971) has given an essentially complete
solution. Among the many results of Brown (1971), the following result,
Theorem 3.6.1 seems to be the most often quoted.

> Theorem Suppose $\Pi$ is spherically symmetric. Hence
> $m_{\Pi}(x)=m_{\Pi}(\|x\|^{2})$. If
>
>
> $\displaystyle\int_{1}^{\infty}\frac{\mathrm{d}t}{t^{d/2}m_{\Pi}(t)}<\infty,$
> (1.5)
>
> then $\delta_{\Pi}$ is inadmissible. If the integral is infinite and the
> risk of $\delta_{\Pi}$ is bounded or equivalently
>
> $\displaystyle\sup_{\mu}E[\|\nabla_{x}\log
> m_{\Pi}(\|X\|^{2})\|^{2}]<\infty,$ (1.6)
>
> then $\delta_{\Pi}$ is admissible.

In Brown (1971), however, this statistically important result is derived as a
corollary of quite deep mathematical results, which many find difficult to
follow. Brown and Hwang (1982) provide a sufficient condition for
admissibility of generalized Bayes estimators, which is much more readable
than Brown (1971) and is based directly on the prior $\pi(\mu)$. However, the
sufficient condition for admissibility is a proper subset of Brown’s. For
example, some important priors including Stein (1974)’s prior

$\displaystyle\pi_{S}(\mu)=\|\mu\|^{2-d}$ (1.7)

which do satisfy Brown’s conditions do not satisfy Brown-Hwang’s. Maruyama and
Takemura (2008) and Maruyama (2009) somewhat enlarged the class of admissible
generalized Bayes estimators when the prior density of $\mu$ is spherically
symmetric. However, these three papers Brown and Hwang (1982), Maruyama and
Takemura (2008) and Maruyama (2009) do not seriously consider the boundary
between admissibility and inadmissibility, which Brown (1971) does.

In this review paper, we assume the prior density is given by

$\displaystyle\pi(\mu)=\int_{0}^{\infty}g^{-d/2}\exp\left(-\frac{\|\mu\|^{2}}{2g}\right)\Pi(\mathrm{d}g),$
(1.8)

with non-negative measure $\Pi$ on $g$, and are going to determine
admissibility/inadmissibility of generalized Bayes estimators for a certain
subclass of mixture priors in terms of $\Pi$ with a completely self-contained
proof. If $\Pi$ is finite or proper, the corresponding Bayes estimator is
admissible. Thus we are mainly interested in the case of infinite $\Pi$.
However, we do not exclude the case of a finite measure $\Pi$.

By the identity

$\displaystyle\|x-\mu\|^{2}+\frac{\|\mu\|^{2}}{g}=\frac{g+1}{g}\left\|\mu-\frac{g}{g+1}x\right\|^{2}+\frac{\|x\|^{2}}{g+1},$

the marginal likelihood $m_{\pi}$ is

$m_{\pi}(\|x\|^{2})=\int_{\mathbb{R}^{d}}\phi(x-\mu)\pi(\mu)\mathrm{d}\mu=\int_{0}^{\infty}(g+1)^{-d/2}\exp\left(-\frac{\|x\|^{2}}{2(g+1)}\right)\Pi(\mathrm{d}g),$
(1.9)

which is finite for all $x$ if

$\displaystyle\int_{0}^{\infty}\frac{\Pi(\mathrm{d}g)}{(g+1)^{d/2}}<\infty.$
(1.10)

Throughout the paper, we assume the prior $\Pi$ satisfies (1.10). By (1.4),
together with $m_{\pi}$ given in (1.9), the generalized Bayes estimator under
$\pi(\mu)$ given by (1.8) is written as

$\displaystyle\delta_{\pi}=x+\nabla_{x}\log
m_{\pi}(\|x\|^{2})=\left(1-\frac{\int_{0}^{\infty}(g+1)^{-d/2-1}\exp\left(-\frac{\|x\|^{2}}{2(g+1)}\right)\Pi(\mathrm{d}g)}{\int_{0}^{\infty}(g+1)^{-d/2}\exp\left(-\frac{\|x\|^{2}}{2(g+1)}\right)\Pi(\mathrm{d}g)}\right)x.$
(1.11)

In earlier studies, the prior (1.8) with some regular varying mixing density
has been used in order to establish minimaxity or both minimaxity and
admissibility of the estimator (1.11). See Remark 3.2. The class of mixing
priors we consider is the following generalization of such mixing densities.

Suppose $\Pi(\mathrm{d}g)$ in (1.8) has a regularly varying density of the
form

$\displaystyle\pi(g;a,b,c)=(g+1)^{a}\left(\frac{g}{g+1}\right)^{b}\left\\{\log(g+1)+1\right\\}^{c},$
(1.12) $\displaystyle\text{for }a<d/2-1,\quad b>-1,\quad c\in\mathbb{R},$
(1.13)

where (1.13) is necessary and sufficient for (1.10). A Tauberian theorem (see,
e.g., Theorem 13.5.4 in Feller (1971)) gives

$\displaystyle\lim_{t\to\infty}\frac{t^{d/2-1}m_{\pi}(t;a,b,c)}{\pi(t;a,b,c)}=\Gamma(d/2-1-a)2^{d/2-1-a}$
(1.14)

where

$\displaystyle
m_{\pi}(\|x\|^{2};a,b,c)=\int_{0}^{\infty}(g+1)^{-d/2}\exp\left(-\frac{\|x\|^{2}}{2(g+1)}\right)\pi(g;a,b,c)\mathrm{d}g.$
(1.15)

Hence the integrability (or non-integrability) of (1.5) of is equivalent to
integrability (or non-integrability) of

$\displaystyle\int_{1}^{\infty}\frac{\mathrm{d}g}{g\pi(g;a,b,c)},$ (1.16)

where, as in Lemma B.2,

$\displaystyle\int_{1}^{\infty}\frac{\mathrm{d}g}{g\pi(g;a,b,c)}\begin{cases}<\infty&\text{either
}a>0\text{ or }\\{a=0\text{ and }c>1\\},\\\ =\infty&\text{either }a<0\text{ or
}\\{a=0\text{ and }c\leq 1\\}.\end{cases}$

Further the risk of the corresponding (generalized) Bayes estimator is bounded
since

$\displaystyle\sup_{x}\|\nabla_{x}\log m_{\pi}(\|x\|^{2};a,b,c)\|^{2}<\infty,$
(1.17)

which is shown in Lemma B.1. Hence, given the result of Brown (1971), we have
a following result.

######  Theorem 1.1.

Suppose the prior $\Pi(g)$ has a regularly varying density, $\pi(g;a,b,c)$
given by (1.12). Then admissibility/inadmissibility of the the corresponding
(generalized) Bayes estimator with $\pi(g;a,b,c)$ is determined by non-
integrability/integrability of

$\displaystyle\int_{1}^{\infty}\frac{\mathrm{d}g}{g\pi(g;a,b,c)},$ (1.18)

or equivalently, of (1.5).

In this paper, we will provide a self-contained proof of this result. Further,
our more general sufficient condition for admissibility in Theorem 3.1,

$\displaystyle\int_{0}^{\infty}\frac{\Pi(\mathrm{d}g)}{g+1}<\infty$ (1.19)

includes the case where the risk of the estimator is not bounded. See Remark
3.1.

The organization of this paper is as follows. Section 2 deals with
inadmissibility and follows a technique used in Dasgupta and Strawderman
(1997). Section 3 is concerned with admissibility. An appendix is devoted to
technical results needed in the development.

##  2 Inadmissibility

For the inadmissibility part of Brown’s theorem, the following proof is
essentially due to Dasgupta and Strawderman (1997), which relates
inadmissibility to solving Riccati differential equations. Following Brown
(1971) and Dasgupta and Strawderman (1997), we do not assume (1.8) but just
spherical symmetry on $\Pi$. Hence the result is presented in terms of
$m_{\Pi}$ (1.1) and $\delta_{\Pi}$ (1.4), not $m_{\pi}$ (1.9) and
$\delta_{\pi}$ (1.11).

######  Theorem 2.1.

The generalized Bayes estimator $\delta_{\Pi}$ under the spherically symmetric
prior $\Pi(\mathrm{d}\mu)$ given by (1.4), is inadmissible if

$\displaystyle\int_{1}^{\infty}\frac{\mathrm{d}t}{t^{d/2}m_{\Pi}(t)}<\infty,$
(2.1)

where $m_{\Pi}(\|x\|^{2}):=m_{\Pi}(x)$.

Notice that the statement of Theorem 2.1 is equivalent to the inadmissibility
condition of Brown (1971). Hence, by (1.14), the inadmissibility part of
Theorem 1.1 follows. More concretely, by Lemma B.2, the integrability (1.18)
corresponds to the case either $a>0$ or {$a=0$ and $c>1$} and hence we have a
following corollary.

######  Corollary 2.1.

The generalized Bayes estimator with the mixing density $\pi(g;a,b,c)$ is
inadmissible if either $a>0$ or {$a=0$ and $c>1$}. For these values of $a$ and
$c$, the integral (1.5) and

$\displaystyle\int_{1}^{\infty}\frac{\mathrm{d}g}{g\pi(g;a,b,c)}$ (2.2)

converges.

###### Proof of Theorem 2.1.

By the Stein (1974) identity, the risk function of an estimator of the form

$\displaystyle\delta_{h}(x)=x+h(x)=(x_{1}+h_{1}(x),\dots,x_{d}+h_{d}(x))^{\mathrm{\scriptscriptstyle
T}}$ (2.3)

is given by

$\displaystyle R(\mu,\delta_{h})$
$\displaystyle=E\left[\|\delta_{h}(X)-\mu\|^{2}\right]$
$\displaystyle=E\left[\|X-\mu\|^{2}\right]+E_{\mu}\left[\|h(X)\|^{2}\right]+2\sum_{i=1}^{d}E_{\mu}\left[(X-\mu)^{\mathrm{\scriptscriptstyle
T}}h(X)\right]$ $\displaystyle=E\big{[}\hat{R}_{h}(X)\big{]},$

where $\hat{R}_{h}(x)$ is called the SURE (Stein Unbiased Risk Estimate) and
is given by

$\displaystyle\hat{R}_{h}(x)=d+\|h(x)\|^{2}+2\sum_{i=1}^{d}\frac{\partial}{\partial
x_{i}}h_{i}(x).$ (2.4)

With $h(x)=\nabla\log
m_{\Pi}(\|x\|^{2})=2xm^{\prime}_{\Pi}(\|x\|^{2})/m_{\Pi}(\|x\|^{2})$ in (2.4),
the SURE of $\delta_{\Pi}$ given by (1.4), is

$\hat{R}_{\Pi}(\|x\|^{2})=d-4w\left(\frac{m^{\prime}_{\Pi}(w)}{m_{\Pi}(w)}\right)^{2}+4d\frac{m^{\prime}_{\Pi}(w)}{m_{\Pi}(w)}+8w\frac{m^{\prime\prime}_{\Pi}(w)}{m_{\Pi}(w)},$
(2.5)

where $w=\|x\|^{2}$. Similarly the SURE of the estimator

$\displaystyle\delta_{\Pi,k}=\delta_{\Pi}-2\frac{k(\|x\|^{2})}{m_{\Pi}(\|x\|^{2})}x$
(2.6)

is

$\displaystyle\hat{R}_{\Pi,k}(\|x\|^{2})=\hat{R}_{\Pi}(\|x\|^{2})+\Delta(\|x\|^{2})$
(2.7)

where

$\displaystyle\Delta(w)=4w\frac{k^{2}(w)}{m_{\Pi}(w)}\left(\frac{1}{m_{\Pi}(w)}-\frac{dk(w)+2wk^{\prime}(w)}{wk^{2}(w)}\right).$
(2.8)

Now let $q(w)=w^{d/2}k(w)$. Then

$\displaystyle\frac{1}{m_{\Pi}(w)}-\frac{dk(w)+2wk^{\prime}(w)}{wk^{2}(w)}=2w^{d/2}\left\\{\frac{1}{2w^{d/2}m_{\Pi}(w)}+\frac{\mathrm{d}}{\mathrm{d}w}\left(\frac{1}{q(w)}\right)\right\\}.$
(2.9)

Assume

$\displaystyle
c=\int_{1}^{\infty}\frac{\mathrm{d}t}{t^{d/2}m_{\Pi}(t)}<\infty.$ (2.10)

Then a solution of the differential equation $\Delta(w)=0$ is given by

$\displaystyle\frac{1}{q_{*}(w)}=-\frac{1}{2}\int_{1}^{w}\frac{\mathrm{d}t}{t^{d/2}m_{\Pi}(t)}+c.$
(2.11)

Hence, under (2.10), the estimator

$\displaystyle\delta_{\Pi,k_{*}}=\delta_{\Pi}-2\frac{k_{*}(\|x\|^{2})}{m_{\Pi}(\|x\|^{2})}x$
(2.12)

with $k_{*}(w)=w^{-d/2}q_{*}(w)$ has the same risk as that of $\delta_{\Pi}$.
Since quadratic loss is strictly convex, the estimator given by the average of
$\delta_{\Pi}$ and $\delta_{\Pi,k_{*}}$,

$\displaystyle\delta_{\Pi}-\frac{k_{*}(\|x\|^{2})}{m_{\Pi}(\|x\|^{2})}x,$
(2.13)

strictly improves on $\delta_{\Pi}$. ∎

######  Remark 2.1.

Note

$\displaystyle\lim_{w\to 0}wk_{*}(w)$ $\displaystyle=\lim_{w\to
0}\frac{w^{-d/2+1}}{-(1/2)\int_{1}^{w}\mathrm{d}t/\\{t^{d/2}m_{\Pi}(t)\\}+c}$
(2.14) $\displaystyle=\lim_{w\to
0}\frac{(-d/2+1)w^{-d/2}}{-1/\\{2w^{d/2}m_{\Pi}(w)\\}}$ (2.15)
$\displaystyle=(d-2)m_{\Pi}(0).$ (2.16)

Then the shrinkage factor of
$\delta_{\Pi}-\\{k_{*}(\|x\|^{2})/m_{\Pi}(\|x\|^{2})\\}x$, defined by

$\displaystyle
1+2\frac{m^{\prime}_{\Pi}(\|x\|^{2})}{m_{\Pi}(\|x\|^{2})}-\frac{k_{*}(\|x\|^{2})}{m_{\Pi}(\|x\|^{2})},$
(2.17)

approaches $-\infty$ as $\|x\|^{2}\to 0$, which implies that the average
estimator (2.13) is dominated by its positive-part estimator. Hence the
average estimator improves on $\delta_{\Pi}$, but is still inadmissible.

Generally speaking, for a generalized Bayes and inadmissible estimator
$\delta_{\Pi}$, it is difficult to find an admissible estimator which
dominates $\delta_{\Pi}$. Interestingly, on page 863 of Brown (1971), there is
some related discussion on this topic.

##  3 Admissibility

Admissibility of (generalized) Bayes estimators is covered in this section.
Subsection 3.1 gives the form of Blyth’s method used, while Subsection 3.2
gives an admissibility result under general $\Pi(\mathrm{d}g)$. Subsection 3.3
considers slowly varying mixing density, $\pi(g;a,b,c)$ with $a=0$, in order
to study the boundary between admissibility and inadmissibility.

###  3.1 Blyth’s method

In Subsection 3.2 a form of Blyth’s method (Blyth (1951)), given in Lemma 3.1
below, is applied to establish admissibility of a class of generalized Bayes
estimators. It utilizes a sequence of proper priors of the form

$\displaystyle\pi_{i}(\mu)=\int_{0}^{\infty}g^{-d/2}\exp\left(-\frac{\|\mu\|^{2}}{2g}\right)h^{2}_{i}(g)\Pi(\mathrm{d}g)$

where $h_{i}(g)$ satisfies
$\int_{0}^{\infty}h^{2}_{i}(g)\Pi(\mathrm{d}g)<\infty$ for any fixed $i$ and
$\lim_{i\to\infty}h_{i}(g)=1$ for any fixed $g$. Specific choice of $h_{i}$
will be given by (3.9) in Section 3.2 and (A.2) in Appendix A.

Under the prior $\pi_{i}(\mu)$, we have

$\displaystyle
m_{i}(\|x\|^{2})=\int_{\mathbb{R}^{d}}\phi(x-\mu)\pi_{i}(\mu)\mathrm{d}\mu=\int_{0}^{\infty}(g+1)^{-d/2}\exp\left(-\frac{\|x\|^{2}}{2(g+1)}\right)h^{2}_{i}(g)\Pi(\mathrm{d}g)$

and

$\delta_{i}=x+\nabla_{x}\log
m_{i}(\|x\|^{2})=\left(1-\frac{\int_{0}^{\infty}(g+1)^{-d/2-1}\exp\left(-\frac{\|x\|^{2}}{2(g+1)}\right)h^{2}_{i}(g)\Pi(\mathrm{d}g)}{\int_{0}^{\infty}(g+1)^{-d/2}\exp\left(-\frac{\|x\|^{2}}{2(g+1)}\right)h^{2}_{i}(g)\Pi(\mathrm{d}g)}\right)x.$
(3.1)

The Bayes risk difference between $\delta_{\pi}$ and $\delta_{i}$, with
respect to $\pi_{i}(\mu)$, is

$\displaystyle\Delta_{i}=\int\left\\{R(\delta_{\pi},\mu)-R(\delta_{i},\mu)\right\\}\pi_{i}(\mu)\mathrm{d}\mu,$

which is rewritten as

$\displaystyle\Delta_{i}$
$\displaystyle=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\left\\{\|\delta_{\pi}-\mu\|^{2}-\|\delta_{i}-\mu\|^{2}\right\\}\phi(x-\mu)\pi_{i}(\mu)\mathrm{d}\mu\mathrm{d}x$
(3.2)
$\displaystyle=\int_{\mathbb{R}^{d}}\bigg{\\{}\big{(}\|\delta_{\pi}\|^{2}-\|\delta_{i}\|^{2}\big{)}m_{i}(\|x\|^{2})-2(\delta_{\pi}-\delta_{i})^{\mathrm{\scriptscriptstyle
T}}\int_{\mathbb{R}^{d}}\mu\phi(x-\mu)\pi_{i}(\mu)\mathrm{d}\mu\bigg{\\}}\mathrm{d}x$
(3.3)
$\displaystyle=\int_{\mathbb{R}^{d}}\|\delta_{\pi}-\delta_{i}\|^{2}m_{i}(\|x\|^{2})\mathrm{d}x.$
(3.4)

The following form of Blyth’s sufficient condition shows that
$\lim_{i\to\infty}\Delta_{i}=0$ implies admissibility. The result applies to
general sequences of prior densities under the given conditions and not just
those constructed as above.

######  Lemma 3.1.

Suppose $\pi_{i}(\mu)$ is an increasing (in $i$) sequence of proper priors,
$\lim_{i\to\infty}\pi_{i}(\mu)=\pi(\mu)$ and $\pi_{i}(\mu)>0$ for all $\mu$.
Then $\delta_{\pi}$ is admissible if $\Delta_{i}$ satisfies
$\lim_{i\to\infty}\Delta_{i}=0$.

###### Proof.

Suppose that $\delta_{\pi}$ is inadmissible and hence that there exists a
$\delta^{\prime}$ satisfies

$R(\mu,\delta^{\prime})\leq R(\mu,\delta_{\pi})$ (3.5)

for all $\mu$ and

$R(\mu,\delta^{\prime})<R(\mu,\delta_{\pi})\text{ for some }\mu_{0}.$ (3.6)

By (3.6), we have

$\displaystyle\int_{\mathbb{R}^{d}}\|\delta_{\pi}(x)-\delta^{\prime}(x)\|^{2}\phi(x-\mu_{0})\mathrm{d}x>0.$

Further we have

$\displaystyle\int_{\mathbb{R}^{d}}\|\delta_{\pi}(x)-\delta^{\prime}(x)\|^{2}\phi(x-\mu)\mathrm{d}x=\int_{\mathbb{R}^{d}}\|\delta_{\pi}(x)-\delta^{\prime}(x)\|^{2}\frac{\phi(x-\mu)}{\phi(x-\mu_{0})}\phi(x-\mu_{0})\mathrm{d}x.$

Since the ratio $\phi(x-\mu)/\phi(x-\mu_{0})$ is continuous in $x$ and
positive, it follows that

$\displaystyle\int_{\mathbb{R}^{d}}\|\delta_{\pi}(x)-\delta^{\prime}(x)\|^{2}\phi(x-\mu)\mathrm{d}x>0$

for all $\mu$.

Set $\delta^{\prime\prime}=(\delta_{\pi}+\delta^{\prime})/2$. Then we have

$\displaystyle\|\delta^{\prime\prime}-\mu\|^{2}=\frac{\|\delta_{\pi}-\mu\|^{2}+\|\delta^{\prime}-\mu\|^{2}}{2}-\frac{\|\delta_{\pi}-\delta^{\prime}\|^{2}}{4},$

and

$\displaystyle R(\mu,\delta^{\prime\prime})$
$\displaystyle=E\left(\|\delta^{\prime\prime}-\mu\|^{2}\right)$
$\displaystyle<(1/2)E\left(\|\delta^{\prime}-\mu\|^{2}\right)+(1/2)E\left(\|\delta_{\pi}-\mu\|^{2}\right)$
$\displaystyle=\frac{1}{2}\left\\{R(\mu,\delta^{\prime})+R(\mu,\delta_{\pi})\right\\}$
$\displaystyle\leq R(\mu,\delta_{\pi}),$

for all $\mu$. Then we have

$\displaystyle\Delta_{i}$
$\displaystyle=\int_{\mathbb{R}^{d}}\left\\{R(\mu,\delta_{\pi})-R(\mu,\delta_{i})\right\\}\pi_{i}(\mu)\mathrm{d}\mu$
$\displaystyle\geq\int_{\mathbb{R}^{d}}\left\\{R(\mu,\delta_{\pi})-R(\mu,\delta^{\prime\prime})\right\\}\pi_{i}(\mu)\mathrm{d}\mu$
$\displaystyle\geq\int_{\mathbb{R}^{d}}\left\\{R(\mu,\delta_{\pi})-R(\mu,\delta^{\prime\prime})\right\\}\pi_{1}(\mu)\mathrm{d}\mu$
$\displaystyle>0$

which contradicts $\Delta_{i}\to 0$ as $i\to\infty$. ∎

Note that the integrand of $\Delta_{i}$ in (3.4) tends to $0$ as $i$ tends to
infinity. Hence, in using the lemma to show admissibility, the bulk of the
remainder of the proof consists in showing that the integrand is bounded by an
integrable function. Then, by the dominated convergence theorem,
$\lim_{i\to\infty}\Delta_{i}=0$ is satisfied so that $\delta_{\pi}$ is
admissible. Establishment of this dominating function constitutes much of the
technical development for the admissibility results given in Sections 3.2 and
3.3.

###  3.2 A general admissibility result for mixture priors

This subsection is devoted to establishing the following result.

######  Theorem 3.1.

The (generalized) Bayes estimator $\delta_{\pi}$ given by (1.11) is admissible
if

$\displaystyle\int_{0}^{\infty}\frac{\Pi(\mathrm{d}g)}{g+1}<\infty.$ (3.7)

Clearly any proper prior on $g$ satisfies (3.7). Further, even if the prior is
improper, i.e., $\int_{0}^{\infty}\Pi(\mathrm{d}g)=\infty$, the corresponding
generalized Bayes estimator is admissible under (3.7). Further, by Lemma B.2,
we have a following corollary.

######  Corollary 3.1.

The (generalized) Bayes estimator with mixing density $\pi(g;a,b,c)$ given by
(1.12) is admissible if either $a<0$ or {$a=0$ and $c<-1$}. For these values
of $a$ and $c$, the integral 1.5 and

$\displaystyle\int_{1}^{\infty}\frac{\mathrm{d}g}{g\pi(g;a,b,c)}$ (3.8)

diverges.

###### Proof of Theorem 3.1.

Let

$\displaystyle h_{i}^{2}(g)=\frac{i}{g+i},$ (3.9)

which is increasing in $i$ and is such that $\lim_{i\to\infty}h_{i}(g)=1$, for
any fixed $g$. The prior $\pi_{i}(\mu)$ is proper for any fixed $i$ since the
mixture distribution is proper, because

$\displaystyle\int_{0}^{\infty}h^{2}_{i}(g)\Pi(\mathrm{d}g)\leq
i\int_{0}^{\infty}\frac{\Pi(\mathrm{d}g)}{g+1}<\infty.$ (3.10)

The integrand of $\Delta_{i}$ given in (3.4) is

$\displaystyle\|\delta_{\pi}-\delta_{i}\|^{2}m_{i}(w)$ (3.11)
$\displaystyle=w\left(\frac{\int_{0}^{\infty}(g+1)^{-1}F(w,g)h_{i}^{2}(g)\Pi(\mathrm{d}g)}{\int_{0}^{\infty}F(w,g)h_{i}^{2}(g)\Pi(\mathrm{d}g)}-\frac{\int_{0}^{\infty}(g+1)^{-1}F(w,g)\Pi(\mathrm{d}g)}{\int_{0}^{\infty}F(w,g)\Pi(\mathrm{d}g)}\right)^{2}\int_{0}^{\infty}F(w,g)h_{i}^{2}(g)\Pi(\mathrm{d}g),$
(3.12)

where $w=\|x\|^{2}$ and

$\displaystyle F(w,g)=(g+1)^{-d/2}\exp\left(-\frac{w}{2(g+1)}\right).$ (3.13)

Applying the inequality

$\left(\sum_{i=1}^{k}a_{i}\right)^{2}\leq k\sum_{i=1}^{k}a_{i}^{2},$ (3.14)

for $k\in\mathbb{Z}$ to (3.11), we have

$\displaystyle\|\delta_{\pi}-\delta_{i}\|^{2}m_{i}(w)$ (3.15)
$\displaystyle\leq
2w\left(\frac{\\{\int_{0}^{\infty}(g+1)^{-1}F(w,g)h_{i}^{2}(g)\Pi(\mathrm{d}g)\\}^{2}}{\int_{0}^{\infty}F(w,g)h_{i}^{2}(g)\Pi(\mathrm{d}g)}+\frac{\\{\int_{0}^{\infty}(g+1)^{-1}F(w,g)\Pi(\mathrm{d}g)\\}^{2}}{\int_{0}^{\infty}F(w,g)\Pi(\mathrm{d}g)}\right).$
(3.16)

Further applying the Cauchy-Schwarz inequality to the first and second terms
of (3.16), we have

$\displaystyle\|\delta_{\pi}-\delta_{i}\|^{2}m_{i}(w)\leq
4w\int_{0}^{\infty}\frac{F(w,g)}{(g+1)^{2}}\Pi(\mathrm{d}g).$

This is precisely the bound required in order to apply the dominated
convergence theorem to demonstrate that $\Delta_{i}\to 0$, since

$\displaystyle\int_{\mathbb{R}^{d}}\|\delta_{\pi}-\delta_{i}\|^{2}m_{i}(\|x\|^{2})\mathrm{d}x$
$\displaystyle\leq
4\int_{\mathbb{R}^{d}}\int_{0}^{\infty}\frac{\|x\|^{2}}{(g+1)^{d/2+2}}\exp\left(-\frac{\|x\|^{2}}{2(g+1)}\right)\Pi(\mathrm{d}g)\mathrm{d}x$
$\displaystyle=4\int_{\mathbb{R}^{d}}\|y\|^{2}\exp\left(-\frac{\|y\|^{2}}{2}\right)\mathrm{d}y\int_{0}^{\infty}\frac{\Pi(\mathrm{d}g)}{g+1}$
$\displaystyle<\infty,$

which completes the proof. ∎

######  Remark 3.1.

The prior $\mu\sim N_{d}(0,gI_{d})$ corresponds to the point prior on $g$ in
(1.8). The proper Bayes estimator is

$\displaystyle\frac{g}{g+1}X$

with unbounded risk

$\displaystyle\frac{g^{2}d+\|\mu\|^{2}}{(g+1)^{2}}.$

Theorem 3.1 covers this case whereas (1.6) of Brown (1971)’s result is not
satisfied by the proper Bayes admissible estimator $gX/(g+1)$.

###  3.3 On the boundary between admissibility and inadmissibility

For the class of densities $\pi(g;a,b,c)$, Corollaries 2.1 and 3.1 settle the
issue of admissibility/inadmissibility for all values of $a$ and $c$ except
for the cases {$a=0$ and $|c|\leq 1$}, where, in particular, the case $a=0$,
$b=0$, $c=0$ is corresponding to the Stein (1974) prior given by (1.7) since

$\displaystyle\int_{0}^{\infty}g^{-d/2}\exp\left(-\frac{\|\mu\|^{2}}{2g}\right)\mathrm{d}g=\Gamma(d/2-1)2^{d/2-1}\|\mu\|^{2-d}.$
(3.17)

For the cases {$a=0$ and $|c|\leq 1$}, the non-integrability

$\displaystyle\int_{1}^{\infty}\frac{\mathrm{d}g}{g\pi(g;a,b,c)}=\infty$
(3.18)

follows. Theorem 1.1 shows that these values, near the boundary between
admissibility and inadmissibility correspond to admissibility. Recall that
Theorem 1.1 is a corollary of Brown (1971). Here, we provide a self-contained
proof of the admissibility result.

######  Theorem 3.2.

Assume the measure $\Pi(\mathrm{d}g)$ has the density $\pi(g;a,b,c)$ with
$a=0$, $b\geq 0$ and $|c|\leq 1$. Then the corresponding generalized Bayes
estimator is admissible.

Proof of Theorem 3.2 will be given in Section A of the Appendix. Unlike
Corollaries 2.1 and 3.1, we assume $b\geq 0$ in Theorem 3.2, which is needed
for an integration by parts. See (A.4) in Appendix for the details.

######  Remark 3.2.

Here is a remark on the regular varying mixing density $\pi(g;a,b,c)$ given by
(1.12),

$\displaystyle\pi(g;a,b,c)=(g+1)^{a}\left(\frac{g}{g+1}\right)^{b}\left\\{\log(g+1)+1\right\\}^{c}.$
(3.19)

The term $(g+1)^{a}$ comes from Strawderman (1971) and Berger (1976), whereas
$\\{g/(g+1)\\}^{b}$ comes from Faith (1978). In these three papers, conditions
for minimaxity or both minimaxity and admissibility of the (generalized) Bayes
estimator (1.11) have been established.

In this paper, we introduce the term $\left\\{\log(g+1)+1\right\\}^{c}$,
mainly in order to clarify the boundary between admissibility and
inadmissibility under $a=0$. Additionally, by Theorem 1 of Fourdrinier,
Strawderman and Wells (1998), we have a following result on minimaxity based
on

$\displaystyle(g+1)\frac{\\{\mathrm{d}/\mathrm{d}g\\}\pi(g;a,b,c)}{\pi(g;a,b,c)}=a+\frac{b}{g}+\frac{c}{\log(g+1)+1}.$
(3.20)

######  Theorem 3.3.

Assume $-d/2+1+\max(0,-2c)\leq a<d/2-1$ and $b\geq 0$. Then the (generalized)
Bayes estimator with the mixing density $\pi(g;a,b,c)$ is minimax.

##  4 Concluding remarks

We have revisited Brown’s monumental 1971, paper on admissibility, and have
given an independent proof of one of his main results for the subclass of
scale mixture priors with mixing density of the form

$\displaystyle\pi(g;a,b,c)=(g+1)^{a}\left(\frac{g}{g+1}\right)^{b}\left\\{\log(g+1)+1\right\\}^{c}.$
(4.1)

For this class of densities, we have established that Brown’s condition for
admissibility/inadmissibility, based on the non-integrability/integrability on
$[1,\infty)$ of $1/\\{t^{p/2}m(t)\\}$ (where $m(\|x\|^{2})$ is the spherically
symmetric marginal density) is equivalent to non-integrability/integrability
of $1/\\{g\pi(g)\\}$ on $[1,\infty)$. We established a sharp bound between
admissibility and inadmissibility for this class. We also established some
results, applicable to more general classes of mixing distributions, that may
be of independent interest. In particular, we have shown that integrability of
$\Pi(\mathrm{d}g)/(g+1)$ on $[0,\infty)$ suffices to establish admissibility
for any generalized mixing prior distribution, independently of whether the
resulting generalized Bayes estimator has bounded risk.

##  Appendix A Proof of Theorem 3.2

In this appendix, it is convenient to use the notation

$\displaystyle\pi(g)=\pi(g;0,b,c)=\left(\frac{g}{g+1}\right)^{b}\left\\{\log(g+1)+1\right\\}^{c}.$

###  A.1 $h_{i}$

Let

$\displaystyle L(g)=\log(g+1)+1.$ (A.1)

Then the non-integrability

$\displaystyle\int_{0}^{\infty}\frac{\mathrm{d}g}{(g+1)L(g)}=\int_{0}^{\infty}\frac{\mathrm{d}z}{z+1}=\infty$

follows. Let

$\displaystyle h_{i}(g)=\begin{cases}\displaystyle
1-\frac{\int_{0}^{g}\mathrm{d}t/\\{(t+1)L(t)\\}}{\int_{0}^{i}\mathrm{d}t/\\{(t+1)L(t)\\}}&0<g<i\\\
0&g\geq i.\end{cases}$ (A.2)

For fixed $g$, $h_{i}(g)$ is increasing in $i$ and
$\lim_{i\to\infty}h_{i}(g)=1$. Since $h_{i}(g)=0$ for $g\geq i$,
$\int\pi(g)h_{i}^{2}(g)\mathrm{d}g<\infty$ even if
$\int\pi(g)\mathrm{d}g=\infty$. Note $h_{i}(g)$ is piecewise differentiable as

$\displaystyle
h^{\prime}_{i}(g)=\begin{cases}\displaystyle-\frac{1}{(g+1)L(g)\int_{0}^{i}\mathrm{d}t/\\{(t+1)L(t)\\}}&0<g<i\\\
0&g\geq i.\end{cases}$

Then $h_{i}^{2}(g)$ is continuously differentiable since
$\\{h_{i}^{2}(g)\\}^{\prime}=2h_{i}(g)h^{\prime}_{i}(g)$ and $h_{i}(i)=0$.
Further we have

$\displaystyle\sup_{i}\left|h^{\prime}_{i}(g)\right|\\{(g+1)L(g)\\}=\begin{cases}\displaystyle\frac{1}{\int_{0}^{1}\mathrm{d}t/\\{(t+1)L(t)\\}}&0<g<1\\\
\displaystyle\frac{1}{\int_{0}^{g}\mathrm{d}t/\\{(t+1)L(t)\\}}&g\geq
1,\end{cases}$ (A.3)

which will be used in Section A.4.

###  A.2 re-expression of the risk difference

Integration by parts of the numerator of $\delta_{i}$ given by (3.1) gives

$\displaystyle\int_{0}^{\infty}(g+1)^{-d/2-1}\exp\left(-\frac{w}{2(g+1)}\right)\pi(g)h_{i}^{2}(g)\mathrm{d}g$
(A.4)
$\displaystyle=\frac{2}{w}\left[(g+1)^{-d/2+1}\exp\left(-\frac{w}{2(g+1)}\right)\pi(g)h_{i}^{2}(g)\right]_{0}^{\infty}$
(A.5)
$\displaystyle\quad+\frac{d-2}{w}\int_{0}^{\infty}(g+1)^{-d/2}\exp\left(-\frac{w}{2(g+1)}\right)\pi(g)h_{i}^{2}(g)\mathrm{d}g$
(A.6)
$\displaystyle\quad-\frac{2}{w}\int_{0}^{\infty}(g+1)^{-d/2+1}\exp\left(-\frac{w}{2(g+1)}\right)\pi^{\prime}(g)h_{i}^{2}(g)\mathrm{d}g$
(A.7)
$\displaystyle\quad-\frac{4}{w}\int_{0}^{\infty}(g+1)^{-d/2+1}\exp\left(-\frac{w}{2(g+1)}\right)\pi(g)h_{i}(g)h^{\prime}_{i}(g)\mathrm{d}g$
(A.8)
$\displaystyle=-\frac{2\pi(0)}{w}\exp\left(-\frac{w}{2}\right)+\frac{(d-2)m_{i}(w)}{w}$
(A.9)
$\displaystyle\quad-\frac{2}{w}\int_{0}^{\infty}(g+1)F(w,g)\pi^{\prime}(g)h_{i}^{2}(g)\mathrm{d}g-\frac{4}{w}\int_{0}^{\infty}(g+1)F(w,g)\pi(g)h_{i}(g)h^{\prime}_{i}(g)\mathrm{d}g,$
(A.10)

where $F(w,g)$ is given in (3.13). For $\delta_{\pi}$, let $h_{i}\equiv 1$ in
(A.4). Note

$\displaystyle(g+1)\frac{\pi^{\prime}(g)}{\pi(g)}=\frac{b}{g}+\frac{c}{L(g)},$
(A.11)

where $L(g)$ is given by (A.1). Then $\|\delta_{\pi}-\delta_{i}\|^{2}m_{i}(w)$
given in (3.4) is

$\displaystyle\|\delta_{\pi}-\delta_{i}\|^{2}m_{i}(w)=4\frac{m_{i}(w)}{w}\left\\{\pi(0)A_{1}(w)-2A_{2}(w)+bA_{3}(w)+cA_{4}(w)\right\\}^{2},$
(A.12)

where

$\displaystyle A_{1}(w)$
$\displaystyle=\frac{\exp(-w/2)}{m_{\pi}(w)}-\frac{\exp(-w/2)}{m_{i}(w)},$
(A.13) $\displaystyle A_{2}(w)$
$\displaystyle=\frac{\int_{0}^{\infty}(g+1)F(w,g)\pi(g)h_{i}(g)h^{\prime}_{i}(g)\mathrm{d}g}{m_{i}(w)},$
(A.14) $\displaystyle A_{3}(w)$
$\displaystyle=\frac{\int_{0}^{\infty}g^{-1}F(w,g)\pi(g)\mathrm{d}g}{m_{\pi}(w)}-\frac{\int_{0}^{\infty}g^{-1}F(w,g)\pi(g)h_{i}^{2}(g)\mathrm{d}g}{m_{i}(w)},$
(A.15) $\displaystyle A_{4}(w)$
$\displaystyle=\frac{\int_{0}^{\infty}\\{1/L(g)\\}F(w,g)\pi(g)\mathrm{d}g}{m_{\pi}(w)}-\frac{\int_{0}^{\infty}\\{1/L(g)\\}F(w,g)\pi(g)h_{i}^{2}(g)\mathrm{d}g}{m_{i}(w)}.$
(A.16)

Further, by the inequality (3.14), we have

$\displaystyle\|\delta_{\pi}-\delta_{i}\|^{2}m_{i}(w)\leq
16\frac{m_{i}(w)}{w}\left\\{\pi(0)^{2}A^{2}_{1}(w)+4A^{2}_{2}(w)+b^{2}A^{2}_{3}(w)+c^{2}A^{2}_{4}(w)\right\\}.$

As noted earlier, the proof is completed by proving dominated convergence for
each of these 4 terms.

###  A.3 Dominated convergence for the term involving $A_{1}$

Since $m_{1}(w)\leq m_{i}(w)\leq m_{\pi}(w)$, we have

$\displaystyle
m_{i}(w)A^{2}_{1}(w)\leq\frac{\exp(-w)}{m_{i}(w)}\leq\frac{\exp(-w)}{m_{1}(w)}=\frac{\exp(-w/2)}{\exp(w/2)m_{1}(w)}.$
(A.17)

Then we have

$\displaystyle\exp(w/2)m_{1}(w)$
$\displaystyle=\int_{0}^{\infty}(g+1)^{-d/2}\exp(w/2)\exp\left(-\frac{w}{2(g+1)}\right)\pi(g)h_{1}^{2}(g)\mathrm{d}g$
$\displaystyle=\int_{0}^{\infty}(g+1)^{-d/2}\exp\left(\frac{gw}{2(g+1)}\right)\pi(g)h_{1}^{2}(g)\mathrm{d}g$
$\displaystyle\geq\int_{0}^{\infty}(g+1)^{-d/2}\pi(g)h_{1}^{2}(g)\mathrm{d}g,$

and hence

$\displaystyle
m_{i}(w)A^{2}_{1}(w)\leq\frac{\exp(-w/2)}{\int_{0}^{\infty}(g+1)^{-d/2}\pi(g)h_{1}^{2}(g)\mathrm{d}g}.$
(A.18)

By Part 1 of Lemma B.3, we have

$\displaystyle\int_{\mathbb{R}^{d}}\frac{m_{i}(\|x\|^{2})}{\|x\|^{2}}A^{2}_{1}(\|x\|^{2})\mathrm{d}x\leq\frac{C_{d}}{\int_{0}^{\infty}(g+1)^{-d/2}\pi(g)h_{1}^{2}(g)\mathrm{d}g}<\infty,$
(A.19)

where

$\displaystyle C_{d}=\frac{\pi^{d/2}2^{d/2}}{d-2}.$ (A.20)

###  A.4 Dominated convergence for the term involving $A_{2}$

By the Cauchy-Schwarz inequality,

$\displaystyle\left(\int_{0}^{\infty}(g+1)F(w,g)\pi(g)h_{i}(g)h^{\prime}_{i}(g)\mathrm{d}g\right)^{2}$
$\displaystyle\leq\int_{0}^{\infty}F(w,g)\pi(g)h^{2}_{i}(g)\mathrm{d}g\int_{0}^{\infty}(g+1)^{2}F(w,g)\pi(g)\\{h^{\prime}_{i}(g)\\}^{2}\mathrm{d}g$
$\displaystyle=m_{i}(w)\int_{0}^{\infty}(g+1)^{2}F(w,g)\pi(g)\\{h^{\prime}_{i}(g)\\}^{2}\mathrm{d}g.$

Then,

$\displaystyle m_{i}(w)A^{2}_{2}(w)$
$\displaystyle\leq\int_{0}^{\infty}(g+1)^{2}F(w,g)\pi(g)\\{h^{\prime}_{i}(g)\\}^{2}\mathrm{d}g$
(A.21)
$\displaystyle\leq\int_{0}^{\infty}(g+1)^{2}F(w,g)L(g)\\{h^{\prime}_{i}(g)\\}^{2}\mathrm{d}g,$
(A.22)

where the second inequality follows from the fact

$\displaystyle\pi(g)=\left(\frac{g}{g+1}\right)^{b}\left\\{\log(g+1)+1\right\\}^{c}\leq
L(g)$ (A.23)

for $b\geq 0$ and $|c|\leq 1$. By (A.3) and Part 2 of Lemma B.3, we have

$\displaystyle\frac{1}{C_{d}}\int_{\mathbb{R}^{d}}\frac{m_{i}(\|x\|^{2})}{\|x\|^{2}}A^{2}_{2}(\|x\|^{2})\mathrm{d}x$
(A.24)
$\displaystyle\leq\int_{0}^{\infty}(g+1)L(g)\sup_{i}\\{h^{\prime}_{i}(g)\\}^{2}\mathrm{d}g$
(A.25)
$\displaystyle\leq\frac{1}{\\{\int_{0}^{1}\mathrm{d}t/\\{(t+1)L(t)\\}\\}^{2}}\int_{0}^{1}\frac{\mathrm{d}g}{(g+1)L(g)}+\int_{1}^{\infty}\frac{1}{(g+1)L(g)}\frac{\mathrm{d}g}{\\{\int_{0}^{g}\mathrm{d}t/\\{(t+1)L(t)\\}\\}^{2}}$
(A.26)
$\displaystyle=\frac{1}{\int_{0}^{1}\mathrm{d}t/\\{(t+1)L(t)\\}}+\int_{1}^{\infty}\left\\{\frac{\mathrm{d}}{\mathrm{d}u}\left(-\frac{1}{\int_{0}^{u}\mathrm{d}t/\\{(t+1)L(t)\\}}\right)\right\\}\mathrm{d}u$
(A.27)
$\displaystyle=\frac{2}{\int_{0}^{1}\mathrm{d}t/\\{(t+1)L(t)\\}}<\infty,$
(A.28)

where $C_{d}$ is given by (A.20).

###  A.5 Dominated convergence for the term involving $A_{3}$

Note

$\displaystyle m_{i}(w)A^{2}_{3}(w)\leq
2\left(\frac{\\{\int_{0}^{\infty}g^{-1}F(w,g)\pi(g)\mathrm{d}g\\}^{2}}{m_{\pi}(w)}+\frac{\\{\int_{0}^{\infty}g^{-1}F(w,g)\pi(g)h_{i}^{2}(g)\mathrm{d}g\\}^{2}}{m_{i}(w)}\right).$
(A.29)

By the covariance inequality, we have

$\displaystyle\int_{0}^{\infty}g^{-1}F(w,g)\pi(g)h_{i}^{2}(g)\mathrm{d}g$
(A.30)
$\displaystyle=\int_{0}^{\infty}\left(\frac{g+1}{g}\right)(g+1)^{-p/2-1}\exp\left(-\frac{w}{2(g+1)}\right)\pi(g)h_{i}^{2}(g)\mathrm{d}g$
(A.31)
$\displaystyle\leq\frac{\int_{0}^{\infty}\\{(g+1)/g\\}(g+1)^{-p/2-1}\pi(g)h_{i}^{2}(g)\mathrm{d}g}{\int_{0}^{\infty}(g+1)^{-p/2-1}\pi(g)h_{i}^{2}(g)\mathrm{d}g}\int_{0}^{\infty}\frac{\pi(g)h_{i}^{2}(g)}{(g+1)^{p/2+1}}\exp\left(-\frac{w}{2(g+1)}\right)\mathrm{d}g.$
(A.32)

Hence we have

$\displaystyle m_{i}(w)A^{2}_{3}(w)\leq
2Q\left(\frac{\\{\int_{0}^{\infty}(g+1)^{-1}F(w,g)\pi(g)\mathrm{d}g\\}^{2}}{m_{\pi}(w)}+\frac{\\{\int_{0}^{\infty}(g+1)^{-1}F(w,g)\pi(g)h_{i}^{2}(g)\mathrm{d}g\\}^{2}}{m_{i}(w)}\right),$
(A.33)

where

$\displaystyle
Q=\left(\frac{\int_{0}^{\infty}\\{(g+1)/g\\}(g+1)^{-p/2-1}\pi(g)\mathrm{d}g}{\int_{0}^{\infty}(g+1)^{-p/2-1}\pi(g)h_{1}^{2}(g)\mathrm{d}g}\right)^{2}.$
(A.34)

Applying the Cauchy-Schwarz inequality to (A.33), we have

$\displaystyle m_{i}(w)A^{2}_{3}(w)\leq
4Q\int_{0}^{\infty}(g+1)^{-2}F(w,g)\pi(g)\mathrm{d}g.$ (A.35)

By Part 2 of Lemma B.3 and (A.35), we have

$\displaystyle\frac{1}{C_{d}}\int_{\mathbb{R}^{d}}\frac{m_{i}(\|x\|^{2})}{\|x\|^{2}}A^{2}_{3}(\|x\|^{2})\mathrm{d}x\leq
4Q\int_{0}^{\infty}\frac{\pi(g)}{(g+1)^{3}}\mathrm{d}g<\infty,$ (A.36)

where $C_{d}$ is given by (A.20).

###  A.6 Dominated convergence for the term involving $A_{4}$

Recall

$\displaystyle
A_{4}(w)=\frac{\int_{0}^{\infty}\\{1/L(g)\\}F(w,g)\pi(g)\mathrm{d}g}{m_{\pi}(w)}-\frac{\int_{0}^{\infty}\\{1/L(g)\\}F(w,g)\pi(g)h_{i}^{2}(g)\mathrm{d}g}{m_{i}(w)},$
(A.37)

where

$\displaystyle L(g)=\log(g+1)+1.$

Then we have

$\displaystyle
m_{i}(w)A^{2}_{4}(w)=\frac{1}{m_{i}(w)}\left(\int_{0}^{\infty}F(w,g)\pi(g)h_{i}^{2}(g)\left\\{\frac{1}{L(g)}-\frac{\int_{0}^{\infty}\\{1/L(g)\\}F(w,g)\pi(g)\mathrm{d}g}{m_{\pi}(w)}\right\\}\mathrm{d}g\right)^{2}.$
(A.38)

By the Cauchy-Schwarz inequality, we have

$\displaystyle m_{i}(w)A^{2}_{4}(w)$
$\displaystyle\leq\int_{0}^{\infty}F(w,g)\pi(g)h_{i}^{2}(g)\left\\{\frac{1}{L(g)}-\frac{\int_{0}^{\infty}\\{1/L(g)\\}F(w,g)\pi(g)\mathrm{d}g}{m_{\pi}(w)}\right\\}^{2}\mathrm{d}g$
(A.39)
$\displaystyle\leq\int_{0}^{\infty}F(w,g)\pi(g)\left\\{\frac{1}{L(g)}-\frac{\int_{0}^{\infty}\\{1/L(g)\\}F(w,g)\pi(g)\mathrm{d}g}{m_{\pi}(w)}\right\\}^{2}\mathrm{d}g$
(A.40)
$\displaystyle=\int_{0}^{\infty}F(w,g)\frac{\pi(g)}{L^{2}(g)}\mathrm{d}g-\frac{\\{\int_{0}^{\infty}\\{1/L(g)\\}F(w,g)\pi(g)\mathrm{d}g\\}^{2}}{\int_{0}^{\infty}F(w,g)\pi(g)\mathrm{d}g},$
(A.41)

where the second inequality follows from the fact $h_{i}^{2}(g)\leq 1$.

When $-1\leq c<1$, we have

$\displaystyle
m_{i}(w)A^{2}_{4}(w)\leq\int_{0}^{\infty}F(w,g)\frac{\pi(g)}{L^{2}(g)}\mathrm{d}g\leq\int_{0}^{\infty}\frac{F(w,g)}{L^{2-c}(g)}\mathrm{d}g$

and, by Part 2 of Lemma B.3,

$\displaystyle\frac{1}{C_{d}}\int_{\mathbb{R}^{d}}\frac{m_{i}(\|x\|^{2})}{\|x\|^{2}}A^{2}_{4}(\|x\|^{2})\mathrm{d}x\leq\int_{0}^{\infty}\frac{\mathrm{d}g}{(g+1)L^{2-c}(g)}=\int_{0}^{\infty}\frac{\mathrm{d}z}{(z+1)^{2-c}}=\frac{1}{1-c}.$

When $c=1$, we need a more careful treatment. In (A.41) we have

$\displaystyle\int_{0}^{\infty}\frac{F(w,g)\pi(g)}{L(g)}\mathrm{d}g$
$\displaystyle=\int_{0}^{\infty}(g+1)^{-d/2}\exp\left(-\frac{w}{2(g+1)}\right)\left(1-\frac{1}{g+1}\right)^{b}\mathrm{d}g$
$\displaystyle\geq\int_{0}^{\infty}(g+1)^{-d/2}\exp\left(-\frac{w}{2(g+1)}\right)\left(1-\frac{\max(b,1)}{g+1}\right)\mathrm{d}g$
$\displaystyle=w^{-d/2+1}\int_{0}^{w}t^{d/2-2}\exp(-t/2)\left(1-t\frac{\max(b,1)}{w}\right)\mathrm{d}t$
$\displaystyle\geq
w^{-d/2+1}\left(2^{d/2-1}\Gamma(d/2-1)-\int_{w}^{\infty}t^{d/2-2}\exp(-t/2)\mathrm{d}t-\max(b,1)\frac{2^{d/2}\Gamma(d/2)}{w}\right).$

Hence there exist $Q_{1}>0$ and $w_{1}>\exp(2Q_{1})$ such that

$\displaystyle\int_{0}^{\infty}\frac{F(w,g)\pi(g)}{L(g)}\mathrm{d}g\geq\frac{2^{d/2-1}\Gamma(d/2-1)}{w^{d/2-1}}\left(1-\frac{Q_{1}}{L(w)}\right)\geq\frac{1}{2}\frac{2^{d/2-1}\Gamma(d/2-1)}{w^{d/2-1}},$
(A.42)

for all $w\geq w_{1}$. Further we have

$\displaystyle\int_{0}^{\infty}F(w,g)\pi(g)\mathrm{d}g$ (A.43)
$\displaystyle=\int_{0}^{\infty}(g+1)^{-d/2}\exp\left(-\frac{w}{2(g+1)}\right)\left(1-\frac{1}{g+1}\right)^{b}\\{\log(g+1)+1\\}\mathrm{d}g$
(A.44)
$\displaystyle=\int_{0}^{\infty}(g+1)^{-d/2}\exp\left(-\frac{w}{2(g+1)}\right)\left(1-\frac{1}{g+1}\right)^{b}\\{\log
w-\log\\{w/(g+1)\\}+1\\}\mathrm{d}g$ (A.45) $\displaystyle=(\log
w+1)\int_{0}^{\infty}\frac{F(w,g)\pi(g)}{L(g)}\mathrm{d}g-w^{-d/2+1}\int_{0}^{w}(\log
t)t^{d/2-2}\exp(-t/2)(1-t/w)^{b}\mathrm{d}t$ (A.46) $\displaystyle\leq
L(w)\int_{0}^{\infty}\frac{F(w,g)\pi(g)}{L(g)}\mathrm{d}g+w^{-d/2+1}\int_{0}^{\infty}|\log
t|t^{d/2-2}\exp(-t/2)\mathrm{d}t.$ (A.47)

By (A.42) and (A.47), for all $w\geq w_{1}$, we have

$\displaystyle\frac{\int_{0}^{\infty}F(w,g)\pi(g)\mathrm{d}g}{\int_{0}^{\infty}\\{1/L(g)\\}F(w,g)\pi(g)\mathrm{d}g}\leq
L(w)+Q_{2}$ (A.48)

where

$\displaystyle Q_{2}=2\frac{\int_{0}^{\infty}|\log
t|t^{d/2-2}\exp(-t/2)\mathrm{d}t}{2^{d/2-1}\Gamma(d/2-1)}.$ (A.49)

Further, for all $w\geq\max(w_{1},\exp(Q_{2}))$, we have $Q_{2}/L(w)<1$ and
hence

$\displaystyle
L(w)+Q_{2}=L(w)\left\\{1+\frac{Q_{2}}{L(w)}\right\\}\leq\frac{L(w)}{1-Q_{2}/L(w)}.$
(A.50)

Then, by (A.42), (A.48) and (A.50), we have

$\displaystyle\frac{\\{\int_{0}^{\infty}\\{1/L(g)\\}F(w,g)\pi(g)\mathrm{d}g\\}^{2}}{\int_{0}^{\infty}F(w,g)\pi(g)\mathrm{d}g}\geq\frac{2^{d/2-1}\Gamma(d/2-1)}{w^{d/2-1}L(w)}\left(1-\frac{Q_{1}+Q_{2}}{L(w)}\right).$
(A.51)

Let $w_{2}=\max(w_{1},\exp(Q_{2}))$ and $Q_{3}=Q_{1}+Q_{2}$. Then, by (A.51),
we have

$\displaystyle\frac{1}{C_{d}}\left(\int_{\|x\|^{2}\leq
w_{2}}+\int_{\|x\|^{2}>w_{2}}\right)\frac{1}{\|x\|^{2}}\frac{\\{\int_{0}^{\infty}\\{1/L(g)\\}F(\|x\|^{2},g)\pi(g)\mathrm{d}g\\}^{2}}{\int_{0}^{\infty}F(\|x\|^{2},g)\pi(g)\mathrm{d}g}\mathrm{d}x$
(A.52)
$\displaystyle\geq\frac{1}{C_{d}}\int_{\|x\|^{2}>w_{2}}\frac{1}{\|x\|^{2}}\frac{(\|x\|^{2})^{-d/2+1}2^{d/2-1}\Gamma(d/2-1)}{L(\|x\|^{2})}\left(1-\frac{Q_{3}}{L(\|x\|^{2})}\right)\mathrm{d}x$
(A.53)
$\displaystyle=\int_{w_{2}}^{\infty}\frac{\mathrm{d}g}{gL(g)}-\int_{w_{2}}^{\infty}\frac{Q_{3}\mathrm{d}g}{g\\{L(g)\\}^{2}}.$
(A.54)

By (A.41), (A.54) and Part 2 of Lemma B.3, we have

$\displaystyle\frac{1}{C_{d}}\int_{\mathbb{R}^{d}}\frac{m_{i}(\|x\|^{2})}{\|x\|^{2}}A^{2}_{4}(\|x\|^{2})\mathrm{d}x$
$\displaystyle\leq\int_{0}^{\infty}\frac{\mathrm{d}g}{(g+1)L(g)}-\int_{w_{2}}^{\infty}\frac{\mathrm{d}g}{gL(g)}+\int_{w_{2}}^{\infty}\frac{Q_{3}\mathrm{d}g}{g\\{L(g)\\}^{2}}$
$\displaystyle=\int_{0}^{w_{2}}\frac{\mathrm{d}g}{(g+1)L(g)}-\int_{w_{2}}^{\infty}\frac{\mathrm{d}g}{g(g+1)L(g)}+\int_{w_{2}}^{\infty}\frac{Q_{3}\mathrm{d}g}{g\\{L(g)\\}^{2}}$
$\displaystyle=\int_{0}^{w_{2}}\frac{\mathrm{d}g}{(g+1)L(g)}+\int_{w_{2}}^{\infty}\frac{Q_{3}\mathrm{d}g}{g\\{L(g)\\}^{2}}<\infty.$

##  Appendix B Lemmas

######  Lemma B.1.

$\displaystyle\sup_{x}\|\nabla_{x}\log m_{\pi}(\|x\|^{2};a,b,c)\|^{2}<\infty.$
(B.1)

###### Proof.

By (1.11), let

$\displaystyle f(\|x\|^{2})=\|\nabla_{x}\log
m_{\pi}(\|x\|^{2};a,b,c)\|=\|x\|\frac{\int_{0}^{\infty}(g+1)^{-d/2-1}\exp\left(-\frac{\|x\|^{2}}{2(g+1)}\right)\pi(g;a,b,c)\mathrm{d}g}{\int_{0}^{\infty}(g+1)^{-d/2}\exp\left(-\frac{\|x\|^{2}}{2(g+1)}\right)\pi(g;a,b,c)\mathrm{d}g},$
(B.2)

where

$\displaystyle
m_{\pi}(\|x\|^{2};a,b,c)=\int_{0}^{\infty}(g+1)^{-d/2}\exp\left(-\frac{\|x\|^{2}}{2(g+1)}\right)\pi(g;a,b,c)\mathrm{d}g.$

Clearly $f(0)=0$ since

$\displaystyle
f(0)=0\times\frac{\int_{0}^{\infty}(g+1)^{-d/2-1}\pi(g;a,b,c)\mathrm{d}g}{\int_{0}^{\infty}(g+1)^{-d/2}\pi(g;a,b,c)\mathrm{d}g}=0.$

Note, as in (1.14),

$\displaystyle\lim_{t\to\infty}\frac{t^{d/2-1}m_{\pi}(t;a,b,c)}{\pi(t;a,b,c)}=\Gamma(d/2-1-a)2^{d/2-1-a}.$
(B.3)

Similarly,

$\displaystyle\lim_{t\to\infty}\frac{t^{d/2}\int_{0}^{\infty}(g+1)^{-d/2-1}\exp\left(-\frac{\|x\|^{2}}{2(g+1)}\right)\pi(g;a,b,c)\mathrm{d}g}{\pi(t;a,b,c)}=\Gamma(d/2-a)2^{d/2-a}.$
(B.4)

Hence, by (B.2), (B.3) and (B.4), we have

$\displaystyle\lim_{t\to\infty}t^{1/2}f(t)=d/2-1-a$

which implies that $\lim_{t\to\infty}f(t)=0$. Together with $f(0)=0$, $f(t)$
is bounded. ∎

######  Lemma B.2.

  1. 1.

For either $a>0$ or {$a=0$ and $c>1$},

$\displaystyle\int_{1}^{\infty}\frac{\mathrm{d}g}{g\pi(g;a,b,c)}<\infty.$
(B.5)

  2. 2.

For either $a<0$ or {$a=0$ and $c<-1$},

$\displaystyle\int_{0}^{\infty}\frac{\pi(g;a,b,c)}{g+1}\mathrm{d}g<\infty.$
(B.6)

  3. 3.

For either $a<0$ or {$a=0$ and $c\leq 1$},

$\displaystyle\int_{1}^{\infty}\frac{\mathrm{d}g}{g\pi(g;a,b,c)}=\infty.$
(B.7)

###### Proof.

[Part 1] Let

$\displaystyle c_{*}=\begin{cases}2&a>0\\\ c&a=0\end{cases}$

and

$\displaystyle
f_{1}(g)=\frac{\\{\log(g+1)+1\\}^{c_{*}-c}}{(g+1)^{a}}\left(\frac{g+1}{g}\right)^{b+1},$
(B.8)

which is bounded for $g\in(1,\infty)$, for either $a>0$ or {$a=0$ and $c>1$},
since $f_{1}(1)<\infty$ and $f_{1}(\infty)<\infty$. Then

$\displaystyle\int_{1}^{\infty}\frac{\mathrm{d}g}{g\pi(g;a,b,c)}$
$\displaystyle=\int_{1}^{\infty}\frac{f_{1}(g)\mathrm{d}g}{(g+1)\left\\{\log(g+1)+1\right\\}^{c_{*}}}$
(B.9) $\displaystyle\leq\max_{g\geq
1}f_{1}(g)\int_{1}^{\infty}\frac{\mathrm{d}g}{(g+1)\left\\{\log(g+1)+1\right\\}^{c_{*}}}$
(B.10) $\displaystyle=\max_{g\geq 1}f_{1}(g)\frac{(\log
2+1)^{1-c_{*}}}{c_{*}-1},$ (B.11)

which completes the proof of Part 1.

[Part 2] For $0<g<1$,

$\displaystyle\frac{\pi(g;a,b,c)}{g+1}=\frac{g^{b}\left\\{\log(g+1)+1\right\\}^{c}}{(g+1)^{-a+(b+1)}}\leq\max\\{1,(\log
2+1)^{c}\\}g^{b}.$ (B.12)

For $g\geq 1$, let

$\displaystyle c_{*}=\begin{cases}2&a<0\\\ -c&a=0\end{cases}$

and

$\displaystyle
f_{2}(g)=(g+1)^{a}\left(\frac{g}{g+1}\right)^{b}\\{\log(g+1)+1\\}^{c+c_{*}}.$
(B.13)

which is bounded for $g\in(1,\infty)$ since $f_{1}(1)<\infty$ and
$f_{1}(\infty)<\infty$. Then, for $g\geq 1$, we have

$\displaystyle\frac{\pi(g;a,b,c)}{g+1}=\frac{f_{2}(g)}{(g+1)\\{\log(g+1)+1\\}^{c_{*}}}\leq\frac{\max_{g\geq
1}f_{2}(g)}{(g+1)\\{\log(g+1)+1\\}^{c_{*}}}.$ (B.14)

By (B.12) and (B.14), we have

$\displaystyle\int_{0}^{\infty}\frac{\pi(g;a,b,c)}{g+1}\mathrm{d}g$
$\displaystyle\leq\max\\{1,(\log
2+1)^{c}\\}\int_{0}^{1}g^{b}\mathrm{d}g+\int_{1}^{\infty}\frac{\max_{g\geq
1}f_{2}(g)}{(g+1)\\{\log(g+1)+1\\}^{c_{*}}}\mathrm{d}g$
$\displaystyle=\frac{\max\\{1,(\log 2+1)^{c}\\}}{b+1}+\frac{(\log
2+1)^{1-c_{*}}}{c_{*}-1},$

which completes the proof of Part 2.

[Part 3] Let

$\displaystyle
f_{3}(g)=\frac{\\{\log(g+1)+1\\}^{1-c}}{(g+1)^{a}}\left(\frac{g+1}{g}\right)^{b+1}.$
(B.15)

which is positive and bounded away from $0$, for either $a<0$ or {$a=0$ and
$c\leq 1$}. Then

$\displaystyle\int_{1}^{\infty}\frac{\mathrm{d}g}{g\pi(g;a,b,c)}$
$\displaystyle=\int_{1}^{\infty}\frac{f_{3}(g)\mathrm{d}g}{(g+1)\left\\{\log(g+1)+1\right\\}}$
(B.16) $\displaystyle\geq\min_{g\geq
1}f_{3}(g)\int_{1}^{\infty}\frac{\mathrm{d}g}{(g+1)\left\\{\log(g+1)+1\right\\}}$
(B.17) $\displaystyle=\min_{g\geq
1}f_{3}(g)\left[\log(\\{\log(g+1)+1\\})\right]_{1}^{\infty}$ (B.18)
$\displaystyle=\infty,$ (B.19)

which completes the proof of Part 3.

∎

######  Lemma B.3.

Assume $d\geq 3$.

  1. 1.

For $\alpha>0$,

$\int_{\mathbb{R}^{d}}\frac{1}{\|x\|^{2}}\exp\left(-\frac{\|x\|^{2}}{\alpha}\right)\mathrm{d}x=\frac{2\pi^{d/2}\alpha^{d/2-1}}{d-2}.$
(B.20)

  2. 2.

For $g\geq 0$, let

$\displaystyle F(w,g)=(g+1)^{-d/2}\exp\left(-\frac{w}{2(g+1)}\right).$ (B.21)

Then

$\displaystyle\int_{\mathbb{R}^{d}}\frac{F(\|x\|^{2},g)}{\|x\|^{2}}\mathrm{d}x=\frac{C_{d}}{g+1},\
\text{ where }\ C_{d}=\frac{\pi^{d/2}2^{d/2}}{d-2}.$ (B.22)

###### Proof.

For Part 1, we have

$\begin{split}\int_{\mathbb{R}^{d}}\frac{1}{\|x\|^{2}}\exp\left(-\frac{\|x\|^{2}}{\alpha}\right)\mathrm{d}x&=\frac{\pi^{d/2}}{\Gamma(d/2)}\int_{0}^{\infty}t^{d/2-1-1}\exp(-t/\alpha)\mathrm{d}t\\\
&=\frac{\pi^{d/2}}{\Gamma(d/2)}\Gamma(d/2-1)\alpha^{d/2-1}\\\
&=\frac{2\pi^{d/2}\alpha^{d/2-1}}{d-2}.\end{split}$ (B.23)

Part 2 follows from Part 1. ∎

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# Dissipative reactions with intermediate-energy beams – a novel approach to
populate complex-structure states in rare isotopes

A. Gade  Facility for Rare Isotope Beams, Michigan State University, East
Lansing, Michigan 48824, USA  Department of Physics and Astronomy, Michigan
State University, East Lansing, Michigan 48824, USA  B. A. Brown  Facility for
Rare Isotope Beams, Michigan State University, East Lansing, Michigan 48824,
USA  Department of Physics and Astronomy, Michigan State University, East
Lansing, Michigan 48824, USA  D. Weisshaar  Facility for Rare Isotope Beams,
Michigan State University, East Lansing, Michigan 48824, USA  D. Bazin
Facility for Rare Isotope Beams, Michigan State University, East Lansing,
Michigan 48824, USA  Department of Physics and Astronomy, Michigan State
University, East Lansing, Michigan 48824, USA  K. W. Brown  Facility for Rare
Isotope Beams, Michigan State University, East Lansing, Michigan 48824, USA
Department of Chemistry, Michigan State University, East Lansing, Michigan
48824, USA  R. J. Charity  Department of Chemistry, Washington University, St.
Louis, Missouri 63130, USA  P. Farris  Facility for Rare Isotope Beams,
Michigan State University, East Lansing, Michigan 48824, USA  Department of
Physics and Astronomy, Michigan State University, East Lansing, Michigan
48824, USA  A. M. Hill  Facility for Rare Isotope Beams, Michigan State
University, East Lansing, Michigan 48824, USA  Department of Physics and
Astronomy, Michigan State University, East Lansing, Michigan 48824, USA  J. Li
Facility for Rare Isotope Beams, Michigan State University, East Lansing,
Michigan 48824, USA  B. Longfellow  Present address: Lawrence Livermore
National Laboratory, Livermore, California 94550, USA  Facility for Rare
Isotope Beams, Michigan State University, East Lansing, Michigan 48824, USA
Department of Physics and Astronomy, Michigan State University, East Lansing,
Michigan 48824, USA  D. Rhodes  Present address: TRIUMF, 4004 Wesbrook Mall,
Vancouver, BC V6T 2A3, Canada  Facility for Rare Isotope Beams, Michigan State
University, East Lansing, Michigan 48824, USA  Department of Physics and
Astronomy, Michigan State University, East Lansing, Michigan 48824, USA  W.
Reviol  Physics Division, Argonne National Laboratory, Argonne, Illinois
60439, USA  J. A. Tostevin  Department of Physics, Faculty of Engineering and
Physical Sciences, University of Surrey, Guildford, Surrey GU2 7XH, United
Kingdom

###### Abstract

A novel pathway for the formation of multi-particle-multi-hole ($np-mh$)
excited states in rare isotopes is reported from highly energy- and momentum-
dissipative inelastic-scattering events measured in reactions of an
intermediate-energy beam of 38Ca on a Be target. The negative-parity, complex-
structure final states in 38Ca were observed following the in-beam
$\gamma$-ray spectroscopy of events in the
9Be($\hbox{${}^{38}$Ca},\hbox{${}^{38}$Ca}+\gamma$)X reaction in which the
scattered projectile lost longitudinal momentum of order $\Delta p_{||}=700$
MeV/c. The characteristics of the observed final states are discussed and
found to be consistent with the formation of excited states involving the
rearrangement of multiple nucleons in a single, highly-energetic projectile-
target collision. Unlike the far-less dissipative, surface-grazing reactions
usually exploited for the in-beam $\gamma$-ray spectroscopy of rare isotopes,
these more energetic collisions appear to offer a practical pathway to
nuclear-structure studies of more complex multi-particle configurations in
rare isotopes – final states conventionally thought to be out of reach with
high-luminosity fast-beam-induced reactions.

###### pacs:

Beyond the proof of existence of a rare isotope and the determination of its
ground-state half-life, the energies of excited states are typically the first
observables that become accessible in laboratory experiments. For excited
bound states, depending on their lifetime, prompt or delayed $\gamma$-ray
spectroscopy is frequently used to obtain precise excitation energies from the
measured transition energies [1]. In short-lived rare isotopes, excited states
can be populated efficiently in (direct) nuclear reactions [2] or $\beta$
decay [3], for example, most often exploiting the unique selectivity inherent
to each of these different population pathways. The selectivity of one- and
two-nucleon transfer and knockout reactions, or inelastic scattering [2, 4, 5,
6, 7, 8], often enhances the population of excited states at moderate spin
associated with the single-particle or collective degree of freedom. Here, we
report the novel, complementary in-beam $\gamma$-ray spectroscopy of higher-
spin, negative-parity states in 38Ca, observed to be populated in the
9Be($\hbox{${}^{38}$Ca},\hbox{${}^{38}$Ca}+\gamma$)X inelastic scattering at
high momentum loss. From the peculiar final states observed, we argue that
these complex-structure, projectile excited states are formed by the
rearrangement of multiple nucleons in a single, highly-energetic projectile-
target collision, giving access to multi-particle configurations not expected
to be in reach of high-luminosity fast-beam reactions.

The reaction channel analyzed here is populated in the same experiment as
reported on in Ref. [9] where the focus was on 40Sc produced in the $pn$
pickup reaction onto the 38Ca projectile. Here, we briefly summarize the
experimental scheme below and refer the reader to Ref. [9, 16] for more
details. The 38Ca rare-isotope beam was produced by fragmentation of a stable
40Ca beam, accelerated to 140 MeV/nucleon by the Coupled Cyclotron Facility at
NSCL [10]. The momentum width transported to the experiment was restricted to
$\Delta p/p=0.25$%, resulting in 160,000 38Ca/s impinging upon a
188-mg/cm2-thick 9Be foil located at the target position of the S800
spectrograph [11]. The setting subject of this publication ran for less than
40 hours. The constituents of the incoming beam and the projectile-like
reaction products were identified on an event-by-event basis using the S800
analysis beam line and focal plane with the standard detector systems [12]. As
the magnetic rigidity of the S800 spectrograph was tuned for 36Ca, only part
of the outermost (exponential) low-momentum tail of the reacted 38Ca
distribution was transmitted to the focal plane. Specifically, the S800
momentum acceptance at this setting is $p_{0}\pm 330$ MeV/c, with
$p_{0}=11.222$ GeV/c.

When compared to the parallel momentum distribution of the unreacted 38Ca
passing through the target, having suffered only in-target energy losses
($p_{0}=11.932$ GeV/c), the low-momentum, reacted 38Ca events detected in the
reaction setting have undergone an additional longitudinal momentum loss of
about 700 MeV/c (see Fig. 1). That is, approximately 18 MeV/c per nucleon in
momentum or 5.4 MeV/nucleon in energy. The cross section for finding 38Ca with
such large momentum loss was extracted to be $\sigma(p_{0}\pm
330~{}\mathrm{MeV/c})=3.8(4)$ mb, making these inelastic large-momentum-loss
events rather rare.

Figure 1: Longitudinal momentum distributions of 38Ca passing through the
target and only suffering energy loss (magenta peak) and, on log scale, for
the dissipative setting (inset (a)). Insets (b) and (c) confront the
$\gamma$-ray spectra in coincidence with less than 100,500 38Ca at high
momentum loss (black) and from nearly 179,000 38Ca in the direct setting
(magenta), highlighting a stark difference in excitation probability.

The mid-target energy of 38Ca in the 9Be reaction target was 60.9 MeV/nucleon.
The target was surrounded by GRETINA [13, 14], an array of 48 36-fold
segmented high-purity germanium crystals assembled into modules of four
crystals each, used for prompt $\gamma$-ray detection to tag the final states
of the reaction residues. Signal decomposition was employed to provide the
$\gamma$-ray interaction points. Of these, the location of the interaction
with the largest energy deposition was selected as the first hit entering the
event-by-event Doppler reconstruction of the $\gamma$ rays emitted from the
reaction residues in-flight at about 33% of the speed of light [14].

The event-by-event Doppler reconstructed $\gamma$-ray spectrum obtained in
coincidence with the 38Ca reaction residues detected in the S800 focal plane
at large momentum loss is shown in Fig. 2. Nearest neighbor addback, as
detailed in [14], was used. Of the seven $\gamma$-ray transitions compiled in
[15], those at 2213(5), 1489(5), 489(4) and 3684(8) keV are observed here,
while the transitions at 214(4), 1048(6), 2417(7), 2537(6), 2688(7), and
2758(7) keV are reported for the first time in the present work. This letter
discusses the strongly-populated states. The reader is referred to the
companion paper for details on some other weakly-populated states [16].

Figure 2: Doppler-reconstructed addback $\gamma$-ray spectrum as detected in
coincidence with the scattered 38Ca nuclei that underwent a large momentum
loss. All $\gamma$-ray transitions are labeled by their energy. The inset
magnifies the high-energy region of the spectrum.

To construct the level scheme, $\gamma\gamma$ coincidences are used. Figure 3
shows the coincidence analysis of the low-energy part of the 38Ca spectrum.
From Fig. 3, it is clear that the 1489-keV $\gamma$ ray feeds the 2213-keV
line, the 489-keV transition feeds the level depopulated by the 1489 keV, and
the 214-keV transition lies on top of the level depopulated by the 489-keV
transition. There is evidence for a weak 1048-keV transition being in
coincidence with the 2213 and 1489 keV $\gamma$ rays.

Figure 3: Left: Doppler-corrected $\gamma\gamma$ coincidence spectra obtained
from cuts on the labeled prominent transitions in the $\gamma\gamma$
coincidence matrix. Background was subtracted via a cut of equal width at
slightly higher energy. Coincidence relationships are evident in the panels.
Right: Resulting level scheme. The width of the arrows is proportional to the
$\gamma$-ray intensity of the corresponding transition. The proton separation
energy of $S_{p}=4.54727(22)$ MeV [19] places the second $3^{-}$ state above
the proton separation energy. The $0^{+}_{2}$ state is shown but was not
populated in the present work.

Figure 3 also shows the partial level scheme with the intensities of the
$\gamma$-ray transitions indicated by the arrow widths. These relative
$\gamma$-ray intensities were deduced from the efficiency-corrected peak areas
from the spectrum displayed in Fig. 2. Remarkably, the fourth strongest
$\gamma$ ray, at 214 keV, has not been reported previously. The relative
intensities and a more complete level scheme, including all $\gamma$-ray
transitions observed, is provided in Ref. [16].

The 1489-keV transition in coincidence with the $2^{+}_{1}\rightarrow
0^{+}_{1}$ decay is consistent with the previously reported ($3^{-}$) state at
3702 keV. The 489-keV transition in coincidence with the 3702-keV ($3^{-}$)
state suggests a level at 4191 keV, which is consistent with a previously
reported state at 4194 keV. However, the $J^{\pi}$ assignment proposed in the
literature of ($5^{-})$ [15] is unlikely as the 489-keV transition in our work
is prompt, on the level of a few ps or faster as evident from the good
resolution and absence of a low-energy tail, which – if of $E2$ character –
would indicate a $B(E2;5^{-}\rightarrow 3^{-})$ strength exceeding the
recommended upper limit of 100 W.u. [17]. From comparison with the mirror
nucleus, 38Ar, which has a 4480-keV $4^{-}$ level with a sole transition of
670 keV connecting to the first $3^{-}$ state, resembling the situation
described here, we propose $J^{\pi}=(4^{-})$ for the 4191-keV state in 38Ca.
The new 214-keV transition feeding the ($4^{-}$) level establishes a state at
4405 keV which appears to correspond to the 4586-keV $5^{-}$ level in the 38Ar
mirror whose far-dominant decay is a 106-keV transition to the $4^{-}$. Based
on mirror symmetry, a ($5^{-}$) assignment is proposed here for the 4405-keV
level in 38Ca. This establishes
$(5^{-})\rightarrow(4^{-})\rightarrow(3^{-}_{1})\rightarrow 2^{+}_{1}$ as the
most intense cascade seen following the 38Ca inelastic scattering populated at
large momentum loss.

The next strongest populated level is the $2^{+}_{2}$ state at 3684 keV for
which only the transition to the ground state is observed here. A $0^{+}$
state at 4748(5) keV is claimed in 38Ca from the $(\hbox{${}^{3}$He},n)$
transfer reaction, however, with the suspicion of a doublet [15]. Due to the
transition to the $(3^{-}$) state, a $0^{+}$ assignment is excluded and the
level established here is tentatively assigned ($3^{-}_{2}$), consistent with
the 4877-keV $3^{-}_{2}$ level in the 38Ar mirror, which also decays
predominantly to the $2^{+}_{1}$ and $3^{-}_{1}$ states [15].

It is interesting to explore which low-lying levels have not been observed in
the present experiment. This is, most prominently, the $0^{+}_{2}$ state
reported at 3084 keV which would decay to the first $2^{+}$ state with a
871-keV transition [15]. There is no evidence for an appreciable presence of
that transition in Figs. 2 and 3 (the 871-keV transition would be 13 keV above
the background feature originating from neutron-induced background as
indicated in Fig. 2).

In the following, we discuss the configurations of the states observed. Many
properties of 40Ca and the surrounding nuclei can be interpreted relative to a
doubly-closed shell structure for the ground state of 40Ca with the $sd$ shell
filled and the $pf$ shell empty. The first excited state of 40Ca has
$J^{\pi}=0^{+}$ and is qualitatively associated with a four-particle four-hole
(4p-4h) state relative to the 40Ca closed-shell ground state [18]. We will use
$\Delta$, the number nucleons moved from $sd$ to $fp$ orbitals, to
characterize the structure of the states. In this notation, the 4p-4h states
in 40Ca have $\Delta=4$. (To remove spurious states, the $\Delta$ basis
includes all components associated with the $\Delta\hbar\omega$ basis
constructed in the $0s$-$0p$-$0d1s$-$0f1p$ model space).

In Ref. [20], a Hamiltonian was developed for these pure $\Delta$
configurations. This Hamiltonian served as the starting point for the new
Florida State University (FSU) Hamiltonian for pure $\Delta$ states [21, 22].
The $A=38$, FSU results are compared to experiment in Fig. 4, the overall
agreement with experiment being good. The calculated configurations can be
divided into those with $\Delta=0$ with positive parity (green), those with
$\Delta=1$ with negative parity (blue) and those with $\Delta=2$ with positive
parity (red).

Figure 4: Comparison of the energies of the low-lying states of 38Ca, with the
states observed here labeled, with shell-model calculations using the FSU
$spsdfp$ interaction, and states in 38Ar [15]. In these plots, the length of
the levels indicates the $J$ value and the color positive parity, $\Delta=2$
(red), negative parity, $\Delta=1$ (blue), and $sd$-shell origin, $\Delta=0$
(green).

In the present 38Ca level scheme, the strongest $\gamma$ rays come from the
$2^{+}_{1}$ state, which is predicted to be of $sd$-shell origin, and from
states with $\Delta$=1, including the highest $J^{\pi}=5^{-}$ level possible
for this $\Delta$. The $\gamma$-ray decay of the 2${}^{+}_{2}$ state is also
observed. In the 36Ar$(\hbox{${}^{3}$He},n)$ reaction in [23] this state is
found to have a strong $(f_{7/2})^{2}$ form factor which would come from
$\Delta=2$ configurations in the FSU spectrum. However, the 0${}^{+}_{2}$
state, which also has $\Delta=2$, was not populated.

In the following, we propose a view that puts the populated states within the
context of the observed high-momentum-loss reaction events. From the
approximately 200 MeV of energy loss in the reaction, and given that the
detected 38Ca are largely within laboratory scattering angles of 3-4∘, about
150 MeV must be dissipated in the 9Be nuclei, with a total binding of 58 MeV.
Thus, there must be disintegration of the target nucleus into a number of
energetic fragments. The emerging picture is then one of multiple nucleons
interacting in a single collision with the formation of complex multi-particle
multi-hole configurations, in contrast to the situation in far-less-
dissipative, surface-grazing collisions. We exclude scenarios where a 38Ca
projectile undergoes multiple collisions within the target as an explanation
for the observed cross sections. High-momentum loss events creating $mp$-$nh$
excitations in such a scenario would require a sequence of knockout and/or
pickup processes and such pickup mechanism cross sections are small – with a
typical upper limit of 2 mb at these beam energies [24].

Connecting to the shell-model picture, excitations within the FSU model space
are described by many-body transition densities. In the simplest scenario,
excitation of the $\Delta=1$ negative-parity states involve the $\Delta=0$ to
$\Delta=1$ one-body transition densities (OBTD). The OBTD to those states
observed are all large. The $\Delta=2$, 2${}^{+}_{2}$ state involves the
$\Delta=0$ to $\Delta=2$ two-body transition density (TBTD). The TBTD
connecting the $\Delta=0$ and $\Delta=2$ 0+ wave functions are the same ones
that enter into the Hamiltonian matrix for mixing these two states. We expect
that the microscopic, two-nucleon excitation mechanism should involve an
operator similar to that of the two-body mixing Hamiltonian (e.g. dominated by
pairing). This would explain why excitation of the 0${}^{+}_{2}$ is not
observed – the mixed 0${}^{+}_{1}$ and 0${}^{+}_{2}$ eigenfunctions are
orthogonal with respect to the two-nucleon excitation operator. We note that
in 40Ca($p,t$) [25] the $0^{+}_{2}$ state is only very weakly populated
compared to the $2^{+}_{2}$ state (see Fig. 1 in Ref. [25]).

The events at momentum losses of 600-700 MeV/c, studied here, are also
reminiscent of observations in the work of Podolyak et al. [30]. There, in the
two-neutron knockout from 56Fe to 54Fe at 500 MeV/nucleon, the population of a
10+ isomer of complex structure was observed in the low-momentum tail of the
parallel momentum distribution at about the same absolute momentum loss. The
authors attributed this population to the excitation of the $\Delta$(1232)
resonance at their relativistic beam energies. This mechanism is not available
to our intermediate-energy beams of tens of MeV/nucleon. One may speculate
that the population of the complex-structure state in the two-neutron knockout
from 56Fe is rather due to a simultaneous multi-nucleon rearrangement as
hypothesized here, without evoking quark degrees of freedom and consistent
with the reduction of multi-step processes at their relativistic energies. For
example, population of the $10^{+}$ state could be due to the $\Delta J=6$
excitation of a $(f_{7/2})^{2}$ $6^{+}$ configuration in 56Fe combined with
the removal of two neutrons from the $1p_{3/2}$ and $0f_{7/2}$ orbitals having
$\Delta J\geq 4$.

In Ref. [16], from the high-spin spectroscopy of states up to $J=15/2$ in
39Ca, we argue that such simultaneous multi-nucleon rearrangement is also at
play in intermediate-energy nucleon transfer reactions, such as
9Be($\hbox{${}^{38}$Ca}^{*},\hbox{${}^{39}$Ca}+\gamma$)X. Once again, these
excitations are seen in events in the tail of the longitudinal momentum
distribution at a momentum loss of 600-700 MeV/c.

In the present work, the specific reaction dynamics at play in the observed
large momentum loss collisions are unclear and remain a challenge for future,
more complete and exclusive measurements. Specifically, it would be critical
to detect the dissociation of the 9Be target nuclei in the large-momentum-loss
events and clarify the kinematics of the residues.

While there is much to be discovered about this type of reaction, it is
evident that this presents a new opportunity in the fast-beam regime which
uniquely complements classic low-energy reactions, such as multi-step Coulomb
excitation and multi-nucleon transfer. Fast beams allow for the use thick
targets and capitalize on an increase in $\gamma$-ray yield by a factor of
about 4300 for the specific example of a 188-mg/cm2 9Be target used here vs. a
1-mg/cm2 Pb target often employed for multi-step Coulomb excitation, for
example. Also, strong forward focusing enhances the collection efficiency as
compared to low-energy reactions that fill a larger phase space.

Multi-step Coulomb excitation studies with low-energy rare-isotope beams have
been performed at beam intensities similar to those used here, but have been
limited to a complementary level scheme selectively comprising cascades
connected by strong $E2$ transitions, with at most the first $3^{-}$ state
[26, 7]. We illustrate this with the example of the state-of-the art low-
energy Coulomb excitation of the neighboring Ca isotope 42Ca on Pb [27]. The
measurement was performed at 1 pnA stable-beam intensity for 5 days (resulting
in more than 110,000 times the number of Ca projectiles on target as in the
present measurement) – excited states up to the $4^{+}_{1,2}$ states were
reported with no evidence for any of the negative-parity cross-shell
excitations observed here.

Multi-nucleon transfer, largely limited to stable beams at pnA beam
intensities, is known to populate complex-structure states, however, without
efficiently reaching 38Ca in spite of 40Ca being an often-used beam (see [28]
and references within). When low-energy neutron-rich beams become available at
near stable-beam intensities, multi-nucleon transfer may become an alternative
to access such states in selected neutron-rich nuclei [29]. While it is
interesting to also extend our approach to collective nuclei, it already
promises to be a unique method to probe cross-shell excitations near magic
numbers, elucidating shell evolution in rare isotopes and exploring the
necessary model spaces for a region’s description on the quest for a
predictive model of nuclei.

In conclusion, the in-beam $\gamma$-ray spectroscopy is reported of higher-
spin, complex-structure negative-parity states in 38Ca populated in highly-
dissipative processes induced by a fast 38Ca projectile beam reacting with a
9Be target. This work constitutes the first high-resolution $\gamma$-ray
spectroscopy of 38Ca with a modern HPGe $\gamma$-ray tracking array. The final
states observed in the inelastic scattering,
9Be($\hbox{${}^{38}$Ca},\hbox{${}^{38}$Ca}+\gamma$)X, at large momentum loss
are characterized through their particle-hole character relative to the 40Ca
closed-shell ground state. Excellent agreement is obtained with shell-model
calculations using the FSU cross-shell effective interaction. Based on the
strongly populated negative-parity states and the non-observation of the first
excited $0^{+}_{2}$ state, we propose a consistent picture in which these
multi-particle multi-hole states are formed by simultaneous rearrangement of
multiple nucleons in a single, highly-dissipative collision. These reaction
processes, seen here in the extreme low-momentum tail of 38Ca+9Be inelastic
scattering, identify a new pathway to gain access to excited states not
usually observed in fast-beam induced reactions and likely out of reach for
low-energy reactions.

This work was supported by the U.S. National Science Foundation (NSF) under
Grants No. PHY-1565546 and PHY-2110365, by the DOE National Nuclear Security
Administration through the Nuclear Science and Security Consortium, under
Award No. DE-NA0003180, and by the U.S. Department of Energy, Office of
Science, Office of Nuclear Physics, under Grants No. DE-SC0020451 (MSU) and
DE-FG02-87ER-40316 (WashU) and under Contract No. DE-AC02-06CH11357 (ANL).
GRETINA was funded by the DOE, Office of Science. Operation of the array at
NSCL was supported by the DOE under Grant No. DE-SC0019034. J.A.T.
acknowledges support from the Science and Technology Facilities Council (U.K.)
Grant No. ST/V001108/1.

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