On a criterion of Riemannian distance for singularity and absolute continuity of probability measures

Statistics & Probability Letters 63 (2003) 185 – 195

On a criterion of Riemannian distance for singularity and absolute continuity of probability measures Yoon Tae Kim∗;1 , Kee Won Lee2 Department of Statistics, Hallym University, Chuncheon, Kangwondo 200-702, South Korea Received July 2001; received in revised form January 2003

Abstract Parametric statistical models, with suitable regularity conditions, have a natural Riemannian manifold structure given by the Fisher information metric. This paper is concerned with the Riemannian distance for geometric conditions on singularity or absolute continuity of probability measures which depend on the parameters. c 2003 Elsevier Science B.V. All rights reserved.
Keywords: Riemannian metric; Riemannian distance; Exponential map; Singularity; Absolute continuity; Geodesic

1. Introduction Let X1 ; : : : ; Xn ; : : : be independent observations, but we shall assume that the densities fi (x; ) (with respect to the measure i ) of observation Xi depend on i where  ∈ ⊆ Rm . The variable Xi takes values on a measurable space (R; B) and has the probability distribution Pi ,  ∈ , with density fi (x; ) = dPi = d i . Let {P ;  ∈ } be a family of probability distributions on (R∞ ; B∞ ) for the sequence (X1 ; : : : ; Xn ; : : :) and let Bn := {X1 ; : : : ; Xn }. Let us write P(n) := P |Bn for the restrictions of measures P to Bn , i.e., let P(n) be measures on (Rn ; Bn ). Then the predictable criterion for singularity or absolute continuity of two measures P0 and P on (R∞ ; B∞ ) is    ∞
fi (xi ; ) = ∞: 1 − E 0 P ⊥ P0 if and only if fi (xi ; 0 ) i=1 ∗

Corresponding author. E-mail address: yoonkim@=sher.hallym.ac.kr (Y.T. Kim). 1 Supported (in part) by KOSEF through Statistical Research Center for Complex Systems at Seoul National University. 2 Supported by the Hallym Academy of Sciences at Hallym University in Korea (2001-5-1).

c 2003 Elsevier Science B.V. All rights reserved. 0167-7152/03/$ - see front matter  doi:10.1016/S0167-7152(03)00082-8

186

Y.T. Kim, K.W. Lee / Statistics & Probability Letters 63 (2003) 185 – 195

P P0

if and only if

∞




 1 − E 0

i=1

 fi (xi ; ) ¡ ∞: fi (xi ; 0 )

Let Mn = {P(n) ;  ∈ } be a family of probability measures on (Rn ; Bn ). We de=ne a mapping u: Mn → Rm by u(P(n) ) = . If this function plays the role of a coordinate function, the vector  = ( 1 ; : : : ;  m ) is used as the coordinates in the parameter space and a diEerential structure may be introduced in Mn by this coordinate function. Thus, for each n, Mn is an m-dimensional diEerentiable manifold. In the dominated case, identifying P(n) with f1 (x; ) · · · fn (x; ), where x =(x1 ; : : : ; x n ), the point in Mn can be considered densities. For each n, under some regularity conditions usually required in the statistical geometric theory (e.g. Amari, 1985), the manifold Mn has a natural Riemannian metric structure, given by its Fisher information matrix  n  @l (x; ) @ln (x; ) n ; a; b = 1; : : : ; m: gab () = E @ a @ b  n We denote the Riemannian metric on the tangent space T (M ), ·; · = mab gab () d a ⊗ d b . (n) Then on the Riemannian manifold (M ; ·; · ) there is a unique symmetric connection (Levi–Civita connection) compatible with ·; · . For further details, see Amari (1982, 1985). Kim (2001) found a geometric condition for singularity of measures which depends on the parameters appearing in Hilbert space-valued stochastic diEerential equations (SDE). Let the solution {X (t); 0 6 t 6 T } be observations with joint distribution {P ;  ∈ } on the set of continuous functions taking values in a Hilbert space. If {Xi (t); i = 1; 2; : : : ; 0 6 t 6 T } is a series representation for the process, the =nite-dimensional projections {Xi (t); i = 1; : : : ; n} of the solution induce the measure Pn = P |Fn , where Fn := {X1 ; : : : ; Xn }. He proved that if the square of distance along -geodesic connecting two measures Pn0 and Pn goes to in=nity as n tends to in=nity, then P and P0 are singular. In this paper, we consider the Riemannian distance for a criterion to singularity or absolute continuity oftwo measures P0 and P on (R∞ ; B∞ ). In the following, summation is taken without the symbol for those indices which appear repeatedly in one term.

2. Riemannian distance and main results It is necessary to remember some basic de=nitions and results in the Riemannian geometry (e.g. Chavel, 1984). Given any tangent vector  ∈ Tp (M ), there is a maximal open interval I in R about the origin and a unique geodesic  in M such that  (0) = p and  (0) = . We assume that M is geodesically complete, i.e., I = R. First, we de=ne the exponential map Tp (M ) → M by expp  =  (1). Thus  (t) = expp t. If r˜ is the norm function r˜ = |v| = v; v 1=2 on Tp (M ), then on 1 any normal neighborhood U of p, the radius function r = r˜ ◦ exp− p is smooth except the point p. Let : [a; b] → M be a piecewise smooth curve segment in a Riemannian manifold M . The arc length of  is  b L() = | (s)| ds: a

Y.T. Kim, K.W. Lee / Statistics & Probability Letters 63 (2003) 185 – 195

187

For any point p and q of a connected Riemannian manifold M , the Riemannian distance d(p; q) from p to q is de=ned by inf {L():  ∈ %(p; q)}, where %(p; q) is the set of all piecewise smooth curve segments in M from p to q. Let B(p; ') = {x ∈ M |d(p; x) ¡ '}. For ' ¿ 0 suLciently small, the B(p; ') is normal and the radial geodesic  from p to q ∈ B(p; ') is the unique shortest curve in M from p to q. In particular, L() = r(q) = d(p; q). First, we obtain a power expansion of the inverse of exponential map at 0 . Lemma 2.1. Let (U; u) be a local chart, where u(Pn ) = . Then the kth component of the tangent 1 n n vector exp− P n (P0 +d  ) on TP (M ) is given by 0

0

1 n k (exp− P n (P0 +d  )) 0



@ k 1 k 1 p i j k = d + *ij (0 ) d d + * (0 ) + *ij (0 )*lp (0 ) d i d j d l 2 6 @ul ij @2 1 @ @ k k 3 q i *jlk (0 ) + 6 i *jlp (0 )*pq + (0 ) + 5 p *lq (0 )*ijp (0 ) 72 @u @u @u @u k



+4

@ k p k k s * (0 )*ijp (0 ) + 2*pq (0 )*sl (0 )*ijs (0 ) + 4*ps (0 )*lq (0 )*ijp (0 ) @ul pq

×d i d j d l d q + O( d5 ):

(1)

Proof. We consider a neighborhood V of the origin in T0 Mn such that exp0 : T0 (Mn ) → M is a diEeomorphism of V onto an open subset W of M . Let (U; u) (U ⊆ W) be a local chart such that P0 ∈ U and P0 +d ∈ U, where u(P ) = . When we take the geodesic (t) = exp0 tv, v ∈ V, then the curve (t) = (u1 (t); : : : ; um (t)) satis=es the equation dui du j d 2 uk k =0 + * ((t)) ij dt 2 dt dt

for = 1; : : : ; m

(2)

i with (0) = P0 and (1) = P0 +d , where m3 numbers *jk denote the ChristoEel symbols. DiEerentiating (2) with respect to t and replacing the second derivative of uk by (2), we have i j l d 3 uk k du du du = 0: + ,ijl 3 dt dt dt dt

(3)

Also diEerentiating (3) and replacing the second derivative of uk by (2), we obtain dui duj dul duq d 4 uk k = 0: ((t)) + , ijlq dt 4 dt dt dt dt

(4)

188

Here,

Y.T. Kim, K.W. Lee / Statistics & Probability Letters 63 (2003) 185 – 195

@ p k k * − 2*p(i) *( j)(l) ; @u(i) ( j)(l)

@ 1
p k k k ,ijlq = , − ,p( j)(l) *(i)(q) : 4  @u(q) (i)( j)(l) k ,ijl

1
= 3 



The notation  in the above expression indicates the cyclic permutation. By using Taylor expansions (2)–(4), it is easy to see that uk (t) = 0k +

d k 1 d2 k 1 d3 k 1 d4 k 2 3 u (0)t + u (0)t + u (0)t + u (0)t 4 + O(t 5 ) dt 2 dt 2 3! dt 3 4! dt 4

d k 1 dui (0) duj (0) 2 u (0)t − *ijk ((0)) t dt 2 dt dt

i @ k 1 du (0) duj (0) dul (0) 3 p k − l *ij ((0)) + 2*pl ((0))*ij ((0)) t + 3! @u dt dt dt

i @ 1 du (0) duj (0) dul (0) duq (0) 4 p k k − q ,ijl ((0)) + ,pjl ((0))*iq ((0)) t + O(t 5 ) + 4! @u dt dt dt dt

= 0k +

and hence



@ k 1 k 1 p i j k u˙ (0) = d + *ij (0 ) d d + * (0 ) + *pl (0 )*ij (0 ) d i d j d l 2 6 @ul ij @2 1 @ @ k k 3 q i *jlk (0 ) + 6 i *jlP (0 )*pq + (0 ) + 5 p *lq (0 )*ijp (0 ) 72 @u @u @u @u k

k



@ k k P k + 4 l *pq (0 )*ijp (0 ) + 2*pq (0 )*sl (0 )*ijs (0 ) + 4*ps (0 )*lqs (0 )*ijp (0 ) @u × d i d j d l d q + O( d5 ): 1 Since (0) ˙ = v and v = exp− 0 ((1)), result (1) follows.

Using Lemma 2.1, an expansion of Riemannian distance can be derived. Theorem 2.1. Under the assumptions in Lemma 2.1, we have the following expansion: d(Pn0 ; Pn0 +d ) n (0 )*ijl (0 ) d i d j d k = gijn (0 ) d i d j + gkl 

1 @ l n l k n s l 3gkl (0 )*ij (0 )*pq (0 ) + 4gpl (0 ) + * (0 ) + *ij (0 )*qs (0 ) 12 @uq ij

×d i d j d p d q + O( d5 ):

Y.T. Kim, K.W. Lee / Statistics & Probability Letters 63 (2003) 185 – 195

189

Proof. First we note that 1 −1 k k exp− 0 (P0 +d  ) = (exp0 (P0 +d  )) (@=@u )

and 1 −1 n n 1=2 d(Pn0 ; Pn0 +d ) = exp− P n (P0 +d  ); expP n (P0 +d  ) : 0

0

Using Lemma 2.1 and (@=@ui ); (@=@uj ) = gij , we can obtain the expansion of the Riemannian distance. In the problem of parameter estimation, let us =x 0 ∈ , which represents the true value of the parameter  to be estimated. We assume that X = (X1 ; X2 ; : : : ; ) with the joint distribution P for  = 0 is observed and estimate 0 from X n = (X1 ; : : : ; Xn ) with the joint distribution P(n) for  = 0 . and P(n) . For each n, we consider the Riemannian distance between the neighboring point of P(n) 0 0 Theorem 2.2. Let P(n) be a neighboring point of the restriction of the true distribution P0 to Bn . Suppose that gijn () converges to a continuous gij () on a neighborhood of  = 0 or diverges for some i; j = 1; : : : ; m. If d(P(n) ; P(n) ) → ∞ as n → ∞, two measures P0 and P are singular. 0 ; P(n) ) → ∞ as n → ∞. Conversely if two measures P0 and P are singular, then d(P(n) 0 Proof. Let us set  = 0 + d, where d is an in=nitesimally small line element. We choose a curve : [0; 1] → M as (t) = P(n) . It is obvious that this curve is smooth with (0) = P(n) and 0 0 +t d  (n) 1 m i i i (1) = P0 +d . Working with the coordinates u ((t)); : : : ; u ((t)), where u ((t)) = 0 + t d , by the de=nition of Riemannian distance, we have  1 (n) (n) 2 d(P0 ; P0 +d ) 6 gijn (0 + t d) d i d j dt: 0

Since limn→∞ d(P(n) ; P(n) ) → ∞, it is easy to see, by the monotone convergence theorem, that 0 0 +d   1 lim gijn (0 + t d) d i d j dt = ∞: (5) 0 n→∞

Now we show that limn→∞ gijn (0 ) = ∞ for some i; j = 1; : : : ; m. Suppose that limn→∞ gijn (0 ) = gij (0 ) ¡ ∞ for all i; j = 1; : : : ; m. Since gij () is continuous on a neighborhood of  = 0 , for in=nitesimally close 0 + t d, 0 6 t 6 1, we obtain  1 gij (0 + t d) d i d j dt ¡ ∞: 0

Clearly (5) implies that limn→∞ gijn (0 ) d i d j = ∞. We consider the Hellinger distance



1 (n) (n) (n) (n) 2 dP0 − dP0 +d ; - (P0 ; P ) = 2 %

190

Y.T. Kim, K.W. Lee / Statistics & Probability Letters 63 (2003) 185 – 195

which can be written as   n
fk1=2 (xk ; 0 + d) : E 0 1 − fk1=2 (xk ; 0 ) k=1 We note that on Riemannian manifold Mn = {P(n) ;  ∈ }, in statistical geometry, the metric tensor n gab and the =rst ChristoEel symbols *ijk are given by n 
@ @ n f (x; ) b fj (x; )fj−1 (x; ) d j (x) = gab () (6) a j @ @ j=1 and *ijk ()

 1 @ @ @ @2 l(x; ) + l(x; ) j l(x; ) l(x; ) =E @ i @ j 2 @ i @ @ k   @2 1 @ @ −1 −2 = f(x; )f (x; ) − f(x; ) j f(x; )f (x; ) @ i @ j 2 @ i @  @ f(x; ) d j (x): @ k 

(7)

From (6) and (7), it follows that ; P(n) ) -2 (P(n) 0 0 +d  =2

n 


(fk1=2 (x; 0 + d) − fk1=2 (x; 0 ))2 d k (x)

k=1 n

=

1
2 j=1 n



1
+ 2 j=1 × =

@ @ fj (x; 0 ) b fj (x; 0 )fj−1 (x; 0 ) d j (x) d a d b a @ @  



@2 1 @ @ fj (x; 0 )fj−1 (x; 0 ) − fj (x; 0 ) b fj (x; 0 )fj−2 (x; 0 ) b a a @ @ 2 @ @

@ fj (x; 0 ) d j (x) d a d b d c + O( d4 ) @ c

1 k 1 n gab (0 ) d a d b + *ab (0 ; n)gkc (0 ; n) d a d b d c + O( d4 ): 2 2

(8)

From (8), we have that -2 (P(n) ; P(n) ) → ∞ because limn→∞ gijn (0 ) d i d j = ∞. By Theorem 0 0 +d  2.37 in Jacod and Shiryaev (1987), P0 and P are singular.

Y.T. Kim, K.W. Lee / Statistics & Probability Letters 63 (2003) 185 – 195

191

Suppose that P0 and P0 +d are singular. From Theorem 2.37 in Jacod and Shiryaev (1987), it follows that -2 (P(n) ; P(n) ) → ∞. By Fubini’s Theorem and HQolder inequality, 0 0 +d  -2 (P(n) ; P(n) ) 0 0 +d    =2 62 =

1 2

=

1 2

2 d fj (x; 0 + t d) dt d j (x) 0 dt

2  1 d 1=2 f (x; 0 + t d) d j (x) dt dt j 0  1 @ @ fj (x; 0 + t d) b fj (x; 0 + t d) d a d b fj−1 (x; 0 + t d) d j (x) dt a @ @ 0  1 n gab (0 + t d) dt: 1

0

(9)

; P(n) ) → ∞ as n → ∞, it follows from (9) that Since -2 (P(n) 0 0 +d   1 n lim gab (0 + t d) dt = ∞: n→∞

0

n (0 ) d a d b = ∞. We can apply Lemma 2.1 to obtain d(P(n) ; P(n) ) → ∞ as So limn→∞ gab 0 0 +d  n → ∞.

By Kakutani dichotomy (e.g. Jacod and Shiryaev, 1987), we have the following Corollary. Corollary 2.1. Under the conditions in Theorem 2.2, we have P0 +d P0

if and only if limn→∞ d(P(n) ; P(n) ) ¡ ∞: 0 0 +d 

3. Examples In this section, we give two examples of parameter estimation problems for Hilbert space-valued SDEs (see Kallianpur and Kim, 1998). The =rst example concerns the case of absolute continuity of the measures P generated by the solution of SDE, but a completely diEerent situation occurs in the second example. Here the measures are singular for diEerent ’s, although the measures corresponding to the =nite-dimensional projections are absolutely continuous. 3.1. Absolutely continuous case Let H be a separable Hilbert space with inner product ·; · and norm  · . Let (%; F; P) be a complete probability space with =ltration (Ft ); t ∈ [0; 1]; satisfying the usual conditions. Let (Wt ) be an (Ft )-cylindrical Brownian motion on H .

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We are going to consider the estimation problem for the following H -valued SDE:  dXt = −( 1 L1 + L0 )Xt dt +  2 AXt dt + B dWt for 0 ¡ t 6 T; X0 = 0;

(10)

where  = ( 1 ;  2 ) is a vector of the unknown parameters. 1 1 Let us put L 1 :=  1 L1 + L0 . Here the operators L− and L− are positive bounded self-adjoints 1 0 with discrete spectrum, and have the same system of eigenfunctions. Let {2k } be the eigenfunctions 1 1 1 of L− and L− which constitute a complete orthonormal system(CONS) in H , and let { − 1 0 k } and 1 {3k−1 } be the corresponding eigenvalues respectively. Then L− is a bounded self-adjoint with 1 1 1   1 discrete spectrum such that L 1 2k = 4k 2k , where 4k :=  k + 3k . We assume that Tt := e−L 1 t is a contraction semigroup on H . It will be assumed that (C1) A∗ 2k = ak 2k ; B∗ 2k = bk 2k ; ak ; bk ¿ 0 for all k where A∗ and B∗ are the adjoint of the operator A and B, respectively. (C2) The sequences {ak } and {bk } satisfy ∞
a2 k

k=1

¡∞

k

∞


and

k=1

b2k ¡ ∞: ( k )1=2

(C3) As n → ∞, n = o(1) 3n

and

an = o(1): n

(C4) The parameter set is given by

= {( 1 ;  2 )| 1 ¿ 0 and  1 ¿ c 2 }

where c = sup k

ak ¿ 0: k

(C5) The sequences {3k } and { k } satisfy ∞ 2
k

k=1

3k

¡ ∞:

From assumptions (C1) and (C2), there exists a unique, continuous H -valued solution X to the SDE (10), i.e., X ∈ C([0; T ]; H ), (P-a.s.) (see Bhatt et al., 1993, Theorem 2.7). A direct calculation shows that the Riemannian metric tensor (gijn ()) is given as follows: n () = g11

n n 1
T
2k 2k − 2 ( 1 k + 3k −  2 ak ) 4 ( 1 k + 3k −  2 ak )2 k=1

+

1 4

n
k=1

k=1

1

2

2k e−2( k +3k − ak )T ; ( 1 k + 3k −  2 ak )2

Y.T. Kim, K.W. Lee / Statistics & Probability Letters 63 (2003) 185 – 195

n g12 () = −

n n 1
T
k a k k ak + 1 2 1 2 ( k + 3k −  ak ) 4 ( k + 3k −  2 ak )2 k=1

k=1

n

−

193

1
k ak e−2( k +3k − ak )T ; 4 ( 1 k + 3k −  2 ak )2 1

2

k=1

n () = g22

T 2 +

n
k=1

1 4

n

1
a2k a2k − ( 1 k + 3k −  2 ak ) 4 ( 1 k + 3k −  2 ak )2

n
k=1

k=1

1

2

a2k e−2( k +3k − ak )T : ( 1 k + 3k −  2 ak )2

n To show that limn→∞ d(Pn0 +d ; Pn0 ) ¡ ∞, it suLces to consider the =rst term in g11 (). It follows from (C4) and (C5) that we have

∞
k=1

∞

∞

k=1

k=1



2 2k 2k k = 6 ¡ ∞: ( 1 k + 3k −  2 ak ) k ( 1 + 3k = k −  2 ak = k ) 3k

So ; P(n) ) 6 lim lim d(P(n) 0 0 +d 

n→∞

n→∞



1

0

gijn (0 + t d) d i d j dt ¡ ∞:

3.2. Singular case Now we consider the estimation problem for the SDE given in equation:  dX (t) = − 1 LX (t) dt +  2 AX (t) dt + B dW (t) for 0 ¡ t 6 T; X (0) = 0:

(11)

Here the operator L−1 is bounded self-adjoint with discrete spectrum. Let {2k } be the eigenfunctions of L−1 which constitute a CONS in H and let {4k−1 } be the corresponding eigenvalues. A; B: H → H are continuous operators. Throughout what follows, it will be assumed that (C1) A∗ 2i = ai 2i ; B∗ 2i = bi 2i ; ai ; bi ¿ 0 for all k, where A∗ and B∗ are the adjoint of the operator A and B, respectively. (C2) The sequences {ai } and {bi } satisfy ∞
ai i=1

4i

¡ ∞;

∞
b2i i=1

4i1=2

¡ ∞:

(C3) The parameter set is given by

= {( 1 ;  2 )| 1 ¿ 0 and  1 ¿ c 2 }

where c = sup i

ai : 4i

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Y.T. Kim, K.W. Lee / Statistics & Probability Letters 63 (2003) 185 – 195

Since the measure P(n) is absolutely continuous with respect to P(n) , we can explicitly compute 0 the Fisher information matrix n g11 ()

1 2 n n n 1
4i2 4i2 T
1
4i2 e−2( 4i − ai )T − = + ; 2 i=1 ( 1 4i −  2 ai ) 4 i=1 ( 1 4i −  2 ai )2 4 i=1 (4i −  2 ai )2

n () = g12

n () g22

n n n T
1
4i ai e−2(4i −ai )T 4i ai 4i ai 1
+ ; − 2 i=1 ( 1 4i −  2 ai ) 4 i=1 ( 1 4i −  2 ai )2 4 i=1 (4i −  2 ai )2

1 2 n n n 1
T
1
a2i e−2( 4i − ai )T a2i a2i − = + : 2 i=1 ( 1 4i −  2 ai ) 4 i=1 ( 1 4i −  2 ai )2 4 i=1 ( 1 4i −  2 ai )2

In addition to assumptions (C1) – (C3), we will assume the following: (C4) As n → ∞, the sequences {ai } and {4i } satisfy n  ( ni=1 ai )2 ai − 1 i=1 n n = o(n ) and → d: i=1 4i i=1 4i Using assumption (C4), as n → ∞, we have n g11 ()

n n T
T 2
n = 4i + ai − + O(1); 2 1 i=1 2( 1 )2 i=1 4( 1 )2

n () g12

n T
=− 1 ai + O(1); 2 i=1

n () = O(1): g22

(12)

Using Theorem 2.2 in Section 3, we will show that the measures P and P0 induced by the solution of SDE (11), corresponding to unknown parameters  = ( 1 ;  2 ), are singular. From (12), we have gijn () d i d j     n n n
T
T
n T = 4i + ai − + O(1) (d 1 )2 − 2 ai + O(1) d 1 d 2 2 i=1 2 2 i=1 4 2 2 i=1 + O(1)(d 2 )2 :

(13)

It follows from assumption (C4) and (13) that limn→∞ gijn (0 ) d i d j = ∞. By Theorem 2.1, we have that limn→∞ d(P(n) ; P(n) ) = ∞. Hence the measures P and P0 are singular, where P is a 0 0 +d  neighboring point of P0 . Using the Hellinger distance, Kallianpur and Kim (1998) also proves that P and P0 are singular. In particular, Kim (2001) shows that if the distance along the -geodesic curve goes to in=nity as n tends to in=nity, two measures are singular.

Y.T. Kim, K.W. Lee / Statistics & Probability Letters 63 (2003) 185 – 195

195

References Amari, S.I., 1982. DiEerential geometry of curved exponential families-curvatures and information loss. Ann. Statist. 10, 357–385. Amari, S.I., 1985. DiEerential-geometrical method in statistics. In: Lecture Notes in Statistics, Vol. 28. Springer, New York. Bhatt, A.G., Kallianpur, G, Karandikar, R.L., Xiong, J., 1993. On interacting systems of Hilbert space valued diEusions. Technical Report 373, Department of Statistics, Center for Stochastic Processes, University of North Carolina. Chavel, I., 1984. Eigenvalues in Riemannian Geometry. Academic Press, New York. Jacod, J., Shiryaev, A.N., 1987. Limit Theorems for Stochastic Processes. North-Holland, Amsterdam. Kallianpur, G., Kim, Y.T., 1998. A curious example from statistical diEerential geometry. Theoret. Probab. Appl. 43 (1), 42–62. Kim, Y.T., 2001. A geometric approach to singularity for Hilbert space-valued SDE. Statist. Probab. Lett. 52, 35–45.