Stability of the optimal filter in a hidden Markov model with multiplicative noise

0.18 MB·English

by  Birgit Debrabant

STABILITY OF THE OPTIMAL FILTER IN A HIDDEN
MARKOV MODEL WITH MULTIPLICATIVE NOISE
3
BIRGIT DEBRABANTANDWILHELMSTANNAT
1
0
2

n
a Abstract. We consider a hidden Markov model with multiplicative noise
J
emergingfromstudiesofsoftwarereliability. Weshowthestabilityoftheop-

8 timalfilterw.r.t.generalinitialconditionsinthetotalvariation-andLp-norm
1 and deduce explicitrates. Remarkably, stability turns out to beindependent
oftheergodicbehavior ofthesignal.

]
R
P 1. Introduction
.
h The stability of nonlinear filters is a field of active research, see e.g. [3], the
t introductionof[9]andreferencestherein. However,the majorityofresultsrequires
a
m the signal process to be ergodic or stable in some sense. In addition, most of the
resultsareobtainedforsignalsobservedwithadditivenoise. Thecaseofnonergodic
[
signals and also the case of signals observed with multiplicative noise still remains

1 mostlyopen. Inthiscontext,thepresentarticlestudiesthestabilityoftheoptimal
v
filter in the following hidden Markov model:
8
2 (1) Signal: X = bX W ,
n n 1 n
4 −
4 (2) Observation: Yn = Xn−1Gn, n∈N,
1. where W , n N, are independent identically Beta(α,β)-distributed random vari-
n
0 ables, α,β >∈0, describing the noise incorporated in the unknown signal, b is a
3
positive parameter depending on which the unknownsignalprocess can be ergodic
1
or nonergodic and where G , n N, are independent Γ(1,β)-distributed. Hence,
: n
v the observation Y depends on X∈ via multiplication with the independent noise
n n
i
X Gn. Thus, model (1) and (2) is an example of filtering a signal observed with
multiplicative noise. Note that although to logarithmize leads to a classical linear
r
a model with additive noise, stability cannot be studied immediately with known
methods such as proposed in e.g. [8, 4] since the corresponding noise terms are
rather irregular e.g. neither unimodal nor do they have light tails.
Essentially, the above model appears in [1] as example for models admitting
explicit invariant conditional distributions. In our case this amounts to the fact
that the incorporated assumptions on the distributions of signal and observation
together with a corresponding initial distribution, i.e.
signal: (X /bX X ) Beta(α,β)
n n 1 n 1
− | − ∼
initial distribution: X Γ(λ,q), where q =α+β and λ>0
0
∼
observation: (Y X ) Γ(X ,β)
n n n
| ∼
imply the following explicit updating rules:
2000 Mathematics Subject Classification. 93E11, 93E15,60G35.
Key words and phrases. Stability,Optimalfilter,Multiplicativenoise.
1
2 BIRGITDEBRABANTANDWILHELMSTANNAT
posterior of X : (X Y ) Γ(λ ,q), where Y =Y ,...,Y
n n 1:n n 1:n 1 n
| ∼
prior of X : (X Y ) Γ(λ /b,α)
n+1 n+1 1:n n
| ∼
1-step ahead prediction: (bX /λ Y ) Pearson Type VI
n+1 n 1:n
| ∼
with parameters β and α, cp. [5]
posterior of X : (X Y ) Γ(λ ,q),
n+1 n+1 1:n+1 n+1
| ∼
where λn+1 = λbn +yn+1
To study software reliability, model (1) and (2) was later applied in [6, 2] as
enhancement to the Kalman filter taking into account that failure data tends to
be highly skewed and observational errors are not mainly caused by instrumental
inaccuracies. Thereby, the observables Y can be interpreted as interfailure times
n
of some software. The X play the role of unknown parameters steering their
n
distribution. Tomodelanevolutionofthesoftwaretheparametersevolveaccording
to (1). The value of b is typically unknown and indicates if we have a tendency
of increasing reliability since e. g. E(Yn|Y1:n−1,X0) = 2b−n X0+ jn=−11bjYj
obviously tends to infinity if b<1. (cid:16) (cid:17)
P
Our stability results give the dependence of the optimal filter πy1:n, that is the
n
regular conditional distribution of X given the observations y = (y ) ,
n 1:n k k=1,...,n
on the initial distribution π of X . To cover a wider range of admissible initial
0 0
conditions we extend the assumptions of [1, 2] and suppose the initial distribution
of X to be a mixed Gamma-distribution with the following density
0
(3) π0(x) xq−1 ∞λqe−λxdU0(λ), x>0,
∝
Z0
where q = α+β and U is a probability measure on a compact subset of (0, ).
0
∞
Such a mixture conserves the conjugacy of this distribution, cp. Lemma 6 which
shows that all the posterior distributions are of a similar type.
The pure Gamma case, where U is a Dirac measure is easy to treat and will
0
serveusasanintroductoryexample,seeSection2. Providedthattheunknowntrue
initialconditionπ˜ is absolutelycontinuousw.r.t.theassumedinitialconditionπ
0 0
with a density bounded awayfrom 0, we show stability in the total variationnorm
(almost surely given the observations) and in the L -norm for arbitrary p > 0
p
(expectation w.r.t.the observationsY ,Y ,...) both with explicit geometricrates.
1 2
In Sections 3 and 4 we pass on to the general mixed Gamma initial condition.
UnderassumptionssimilartothoseinSec.2weshowstabilitywithgeometricrates.
Concerning the total variation norm, Theorem 5 gives that almost surely
(4) ||πny1:n −π˜ny1:n||var = O(δ−n), n→∞,
for δ <E(W1β)−β1, which coincides with the pure Gamma case. Note that the rate
of stability 1 is independent of the parameter b in the signal. That is, the filter
δ
is stable whether or not the signal is ergodic. However, the constant on the right
hand side of (4) will depend on the given sequence y ,y ,... of observations. For
1 2
the L -norm with p 0, u0 lnE(Wβ) , Theorem 9 yields
p ∈ −B¯ 1
(cid:16) (cid:17)
E πY1:n π˜Y1:n p = (ρn), n ,
|| n − n ||var O →∞
p
where ρ = E(W1β)e(cid:0)puB0¯ p+β and where(cid:1) B¯ = B¯(β,u0,o0) is a positive constant
specified in (cid:16)the theorem.(cid:17)These rates are smaller compared to ρ = E(Wp) in the
1
pure Gamma case.
STABILITY OF THE OPTIMAL FILTER IN A HMM WITH MULTIPLICATIVE NOISE 3
All stability statements of this article are based on an universal stability result
of [7]. Therein, a general bound for the total variation distance of optimal filters
w.r.t.totheinitialconditionisdeducedintermsoftheLipschitzcontractionχ of
∗n
acertainparabolicgroundstatetransformPn∗ associatedwithπny1:n. Moreprecisely,
suppose that the optimal filter is erroneously initialized with initial condition π
0
and that π˜ is the true initial condition. If π˜ π with density h = dπ˜0 it
fNo,llowwhsertehaPt0n∗fπ˜ny:1=:n π≪ny1:nπgnyR(1·g,:yn(ns,)wyPˆni()tfPhˆππnydny1−1:−en1:nn1−−s11i)t(ys)dhsn, Pgˆivr0eens≪pb.ypˆ0hisnt=hePdn∗ua◦l·o·p·e◦r0aPt1∗ohr0od,fπn0th∈e
transitionprobabilityP resp.thetransitiondensitypofthesignal,thatisP(X
n+1
∈
dx X ) = p(X ,x )dx and Pˆf = p(x, )f(x)dx, and where g denotes
n+1 n n n+1 n+1
| ·
the regular conditional density of the observation given the signal, i.e. P(Y
n
R ∈
dy X ) = g(X ,y)dy. It follows that the error between the true optimal filter
n n
|
π˜y1:n and the erroneously initialized filter πy1:n can be expressed in terms of the
n n
Lipschitz contractions χ of the Markovian transition kernels P , whereby χ :=
∗n n∗ ∗n
P f
sup || n∗ ||Lip and where Lip is the space of all Lipschitz continuous functions
f
f Lip Lip
∈ || ||
with the corresponding norm :
Lip
||·||
Proposition 1 (cp. Prop. 2.1 of [7]). For the total variation distance of the true
optimalfilterπ˜y1:n toawronglyinitialized πy1:n thefollowing explicit boundisvalid:
n n
n
σˆ
(5) ||πny1:n −π˜ny1:n||var ≤ 2Hn · χ∗k·||h0||Lip, n∈N,
k=1
Y
provided that π˜ has π -density h which is bounded away from 0 by a positive
0 0 0
constant H and with σˆn = ∞∞|x1−x2|dπny1:n(x1)dπny1:n(x2).
00
RR
Note that the formulation in [7] is more general and gives a bound also in the
case H =0.
Clearly, for our model (1) and (2) we obtain
(bx)1 q
p(x,y)= Beta(α−,β)yα−1(bx−y)β−11l[0,bx](y) and
(by)1 q
pˆ(x,dy)=p(y,x)dy = Beta(α−,β)xα−1(by−x)β−11l[b−1x,∞)(y)dy.
2. Stability w.r.t. pure Gamma initial conditions
We consider the particular case π (x) xq 1λqe λx, x > 0, for some λ > 0,
0 − −
∝
hence X is Γ(λ,q)-distributed. Using the recursion
0
(6) πy1:n+1(x) g(x,y ) p(s,x)πy1:n(ds),
n+1 ∝ n+1 n
Z
it is straightforward to show that the corresponding optimal filters πy1:n, n N,
are again Gamma-distributed, namely Γ(λ ,q) with λ =b nλ+ nn bj ny∈.
n n − j=1 − j
Tostudytheasymptoticdependenceoftheoptimalfilterontheinitialcondition
P
we apply Prop. 1. Since
(7) p∗n(x,dy) = λΓβn(−β)1(y−b−1x)β−1e−λn−1(y−b−1x)1l(b−1x,∞)(y)dy,
4 BIRGITDEBRABANTANDWILHELMSTANNAT
we find P =b 1, n N. Further
|| n∗||Lip − ∈
1
2
∞∞ √2q
σˆ (x x )2dΓ(λ ,q)(x )dΓ(λ ,q)(x ) = , n N.
n 1 2 n 1 n 2
≤  −  λ ∈
n
ZZ
0 0
 
Prop. 1 then implies
√2q h
(8) ||πny1:n −π˜ny1:n||var ≤ |2|H0||Lip(bnλn)−1, n∈N,
if the true initial condition π˜ has a π -density which is bounded away from 0 by
0 0
some constant H >0.
The following lemma analyses the limiting behavior of bnλ = λ+ n bjy
n j=1 j
which evidently plays a crucial role concerning stability:
P
uLleamr,ma∞j=21.bPjY(jlim=i+nf∞j→a∞lm{bojsYtjsu>reδljy}.)=1 for arbitrary δ <E(W1β)−β1. In partic-
Proof.PWe may assume δ 0. Note that
≥
Xβ δjb−j
P(bjYj δj) = E j yβ−1e−Xjydy
≤ Γ(β)Z0 !
δjβ
(9) E (b jX )β
− j
≤ Γ(β+1) ·
=(E(cid:0)(Wβ))jE(X(cid:1)β)
1 0
| {z }
1
is summable for δ < E Wβ −β. Hence, the Borel-Cantelli lemma yields
1
Pim(plilmiesintfhj→at∞{b∞jj=Y1jb>jYjδjd}i)ve=(cid:16)rg1e.s(cid:17)tSoinicnefinEit(yWa1βlm)−oβ1st>su1r,elwy.e can choose δ > 1 whic(cid:3)h
P
Remark3. Lemma2remainsvalidforanyinitialconditionπ satisfyingE(Xβ)< .
0 0 ∞
Concerningalmostsurestability,Lemma2impliesthat(8)convergesto0almost
surely and behaves as O(δ−n) for every δ <E(W1β)−β1.
Moreover,concerning stability in the Lp-norm, for 0<p<β we find
E((bnYn)−p) = E ∞(bny)−p Xnβ yβ−1e−Xnydy
Γ(β)
(cid:18)Z0 (cid:19)
Γ(β p) Γ(β p)E(Xp)
= Γ(−β) E (b−nXn)p = −Γ(β) 0 E(W1p)n,
(cid:0) (cid:1)
n N. Therefore, the optimal filter is stable in the corresponding Lp-norm and
∈
satisfies
E πY1:n π˜Y1:n p = (ρn), n ,
|| n − n ||var O →∞
with ρ=E(Wp)=(cid:0)Beta(α+p,β). (cid:1)
1 Beta(α,β)
STABILITY OF THE OPTIMAL FILTER IN A HMM WITH MULTIPLICATIVE NOISE 5
3. Stability in the total variation norm
We return to the more general mixed Gamma initial conditions of the form (3)
anddeduceanestimateofthetotalvariationdistanceofπy1:n w.r.t.differinginitial
n
conditionsinThm.4whichthenimpliesalmostsurestabilitywithgeometricrates,
cp. Thm. 5.
Theorem 4. Let π and π˜ be such that
0 0
• π0(x) ∝ xq−1 0∞λqe−λxdU0(λ) for some probability measure U0 with sup-
port [u ,o ] (0, ), and
0 0
π˜ is a proba⊂bRility∞density on R such that the corresponding density h :=
0 + 0
• π˜0 is Lipschitz continuous and h H >0 for some constant H.
π0 0 ≥
Let y1:n ∈ Rn+ and let πny1:n resp. π˜ny1:n be the optimal filter w.r.t. π0 resp. π˜0.
Then
n 1
(10) ||πny1:n −π˜ny1:n||var ≤ bnQu − 1+Bo0bk−uu0 ||h0||Lip, n∈N,
n k=0(cid:18) k (cid:19)
Y
where Q= √q(q+1) and B = 1 β(β+1) 2β2u0 + β(β+1)o20 21.
√2H 2 − o0 u20
(cid:16) (cid:17)
This estimate induces stability with geometric rates:
Theorem 5. Under the assumptions of Theorem 4 the optimal filter is stable for
almost all sequences y ,y ,... of observations and we have
1 2
(11) ||πny1:n −π˜ny1:n||var = O(δ−n), n→∞,
for every δ <E(W1β)−β1.
Proofs and preliminary results. Before proving the above main statements we con-
sider two preliminary results. Firstly note that posterior distributions correspond-
ing to an initial condition of the form (3) are again of mixed Gamma type:
Lemma 6. The posterior distributions πy1:n are mixed Gamma-distributions with
n
densities
πny1:n(x)∝xq−1 ∞λqe−λxdUn(λ), x>0.
Z0
Hereby, the distributions U , n N, obey the following recursive scheme:
n
∈
U(1)(dλ)=λαU (dλ),
n n
•
U(2)(A)=U(1)(b A) for A (R ),
n n +
• · ∈B
U(3) = δ ⋆U(2), where δ denotes the Dirac measure in y and ⋆
• n yn+1 n yn+1 n+1
the convolution,
U (dλ)=λ qU(3)(dλ).
n+1 − n
•
Remark7. ForU0 =δλ with someλ>0weobtain Un =δb−nλ+Pnk=1bk−nyk. This
setup corresponds to the pure Gamma case of Section 2.
If U has compact support [u ,o ] then U has also a compact support [u ,o ]
0 0 0 n n n
with u =b nu + n bk ny and o =b no + n bk ny , n N.
n − 0 k=1 − k n − 0 k=1 − k ∈
P P
6 BIRGITDEBRABANTANDWILHELMSTANNAT
Proof. The recursion (6) implies
πy1:n+1(dx)
n+1
xβe−xyn+1 ∞ (bs)1−qxα−1(bs x)β−1 sq−1 ∞λqe−λsdUn(λ)ds
∝ Zb−1x − · Z0
xq−1e−xyn+1 ∞λαe−λb−1xdUn(λ)
∝
Z0
= xq−1e−xyn+1 ∞e−λb−1xdUn(1)(λ) = xq−1e−xyn+1 ∞e−λxdUn(2)(λ)
Z0 Z0
= xq 1 ∞e λxdU(3)(λ) = xq 1 ∞λqe λxdU (λ), x>0.
− − n − − n+1
Z0 Z0
(cid:3)
In order to apply Prop. 1 we deduce an upper bound for the contraction coeffi-
cient χ of the ground state transform P :
∗n n∗
Proposition 8. Let f : R R be Lipschitz-continuous. Under the assumptions
+
→
of Thm. 4 we find
1sup d (x)+1
||Pn∗f||Lip ≤ 2 x>0bn ||f||Lip
with
1
∞∞ β(β+1) 2β2 β(β+1) 2
d (x) = λ λ + dUx (λ )dUx (λ )
n ZZ | 1− 2|(cid:18) λ21 − λ1λ2 λ22 (cid:19) n−1 1 n−1 2
0 0
and
λαe λb−1xdU (λ)
dUnx−1(λ)= λαe−−λb−1xdUnn−11(λ), n∈N, x>0.
−
Moreover R
1
o 2β2u β(β+1)o2 2
sxu>p0dn(x) ≤ (cid:18)unn−−11 −1(cid:19)(cid:18)β(β+1)− o0 0 + u20 0(cid:19) , n∈N.
Notations: To simplify notations in the following proof we introduce for x,y > 0
and n N the following measures derived from U :
n
∈
λαe λb−1xdU (λ)
dUx(λ) = − n ,
n 0∞λαe−λb−1xdUn(λ)
dUnx,y(λ) = RΓλ(ββ)yβ−1e−λydUnx(λ),
dU¯ (λ,y) = dUx,y(λ)dy.
n n
STABILITY OF THE OPTIMAL FILTER IN A HMM WITH MULTIPLICATIVE NOISE 7
Moreover,note the following estimate via Jensen’s inequality
∞∞
y y dΓ(λ ,β)(y )dΓ(λ ,β)(y )
1 2 1 1 2 2
| − |
ZZ
0 0
1
2
∞∞
(y y )2dΓ(λ ,β)(y )dΓ(λ ,β)(y )
1 2 1 1 2 2
≤  − 
ZZ
0 0
 1 
β(β+1) 2β2 β(β+1) 2
(12) = + .
λ2 − λ λ λ2
(cid:18) 1 1 2 2 (cid:19)
Proof. Let n N. Due to the structure of the optimal filter we obtain for x,y >0
∈
p∗n(x,y) = R0∞Γe(−βλ)y(0y∞−λαbe−−1xλb)−β1−x1dλUqnd−U1n(−λ1)(λ)1l(b−1x,∞)(y).
R
For x ,x >0 we find
1 2
Pn∗f(x1)−Pn∗f(x2)
∞ x1 x2 x2
= f y+ f y+ p x ,y+ dy
b − b ∗n 2 b
Z0 h (cid:16) (cid:17) (cid:16) (cid:17)i (cid:16) (cid:17)
=:T1
+| ∞f y+ x2 p x{z,y+ x1 p x ,y+ x}2 dy.
b ∗n 1 b − ∗n 2 b
Z0 (cid:16) (cid:17)h (cid:16) (cid:17) (cid:16) (cid:17)i
=:T2
| {z }
As x p (x,y+b 1x) is differentiable with
7→ ∗n −
d x
p x,y+
dx ∗n b
= (cid:16)0∞−λb−(cid:17)1e−λ(y+b−1x)yβ−1λqdUn−1(λ)
R 0∞λαe−λb−1xdUn−1(λ)Γ(β)
+ 0∞Re−λ(y+b−1x)yβ−1λqdUn−1(λ) 0∞λb−1λαe−λb−1xdUn−1(λ)
R 0∞λαe−λb−1xdURn−1(λ) 2Γ(β)
= ∞ λdUx,y (λ(cid:0))R+ ∞dUx,y (λ) ∞(cid:1)∞ λdUx,y (λ)dy,
−Z0 b n−1 Z0 n−1 ·Z0 Z0 b n−1
for T we find
2
∞ x2 x2 d x
T2 = f y+ b dxp∗n x,y+ b dxdy
Z0 (cid:16) (cid:17)Zx1 (cid:16) (cid:17)
x2 ∞ x2 d x
= f y+ p x,y+ dydx.
b dx ∗n b
Zx1 Z0 (cid:16) (cid:17) (cid:16) (cid:17)
=:T3
| {z }
8 BIRGITDEBRABANTANDWILHELMSTANNAT
Further we have
∞∞ x λ
T = f y+ 2 dU¯x (λ,y)
3 b b n−1
Z0 Z0 (cid:16) (cid:17)
∞∞ x ∞∞λ
f y+ 2 dU¯x (λ,y) dU¯x (λ,y)
− b n−1 · b n−1
Z0 Z0 (cid:16) (cid:17) Z0 Z0
1 ∞∞∞∞ x x λ λ
2 2 1 2
= f y + f y +
1 2
2 b − b b − b
Z0 Z0 Z0 Z0 (cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17)(cid:18) (cid:19)
dU¯x (λ ,y )dU¯x (λ ,y ),
n 1 1 1 n 1 2 2
− −
since U¯x is a probability measure on [0, )2. Therefore,
n−1 ∞
f ∞∞∞∞
T || ||Lip y y λ λ dU¯x (λ ,y )dU¯x (λ ,y )
| 3| ≤ 2b | 1− 2|·| 1− 2| n−1 1 1 n−1 2 2
ZZZZ
0 0 0 0
f ∞∞∞∞ (λ λ )β
= || 2||bLip |y1−y2|· Γ1(β2)2 (y1y2)β−1e−λ1y1−λ2y2dy1dy2
ZZZZ
0 0 0 0
λ λ dUx (λ )dUx (λ )
·| 1− 2| n−1 1 n−1 2
and formula (12) yields
1
f ∞∞ λ λ 2β2λ β(β+1)λ2 2
T || ||Lip | 1− 2| β(β+1) 1 + 1
| 3| ≤ 2b λ − λ λ2
ZZ 1 (cid:18) 2 2 (cid:19)
0 0
dUx (λ )dUx (λ )
n 1 1 n 1 2
− −
1
f o 2β2u β(β+1)o2 2
|| ||Lip n−1 1 β(β+1) 0 + 0
≤ 2b u − − o u2
(cid:18) n−1 (cid:19)(cid:18) 0 0 (cid:19)
=:2B
since 1≤ uonn ≤ uo00 and uonn ↓1 due|to Lemma 2. For{zT2 we finally hav}e
f o
Lip n 1
T2 x1 x2 || || − 1 B
| | ≤ | − | b u −
(cid:18) n−1 (cid:19)
which together with the obvious estimate |T1|≤ ||f|b|Lip ·|x1−x2| yields
1+ on−1 1 B
P f un−1 − f .
|| n ||Lip ≤ (cid:16) b (cid:17) || ||Lip
(cid:3)
Now we can proceed with the proofs of Theorems 4 and 5:
Proof. of Thm. 4: We apply Prop. 1. Due to Prop. 8 we have
o u o u
χ∗k ≤ b−1 1+B k−1u− k−1 = b−1 1+Bbk01−u 0 .
(cid:18) k−1 (cid:19) (cid:18) − k−1(cid:19)
STABILITY OF THE OPTIMAL FILTER IN A HMM WITH MULTIPLICATIVE NOISE 9
For σˆ we find
n
∞∞∞∞
σˆ = x x dΓ(λ ,q)(x )dΓ(λ ,q)(x )dU (λ )dU (λ )
n 1 2 1 1 2 2 n 1 n 2
| − |
ZZZZ
0 0 0 0
1
2
∞∞q(q+1) q2 q(q+1)
2 + dU (λ )dU (λ )
≤  λ2 − λ λ λ2 n 1 n 2 
ZZ 1 1 2 2
0 0
≤ u−n1(2q(q+1))21 
due to (12). Altogether this yields (10) and proves the theorem. (cid:3)
Proof. of Thm. 5:
We show that (bnun)−1 kn=−01 1+Bob0k−uuk0 vanishes almost surely if n goes to
infinity: (cid:16) (cid:17)
Q
First note that
n 1 n 1
k=−0(cid:18)1+Bo0bk−uku0(cid:19) ≤ exp(B(o0−u0)k=−0(bkuk)−1),
Y X
since 1+x expx for x R.
Rememb≤er that bku =∈u + k bjy . Due to Lemma 2 we canfind δ >1 and
k 0 j=1 j
a random index J a.s. finite such that Y >(δb 1)j for all j J . Consequently,
there is a j N wδith P j − ≥ δ
δ
∈
(13) kn=−01 1+Bob0k−uuk0 exp B¯ jkδ=−01 u10 +B¯ kn=−j1δ δ1k ,
Q (cid:16)bnun (cid:17) ≤ n Pu0+ nj=jδδPj o
where B¯ =B(o u ). The right hand side of (13) cPan be bounded for n j by
0 0 δ
− ≥
exp B¯j exp B¯ δ
(14) u0 δ δ−1 ,
n oδn n o
which tends to 0 as (δ n) for n . (cid:3)
−
O →∞
4. Stability in the Lp-norm
In the setting of Theorem 4 the optimal filter is always stable in the Lp-norm
for every p > 0 since the total variation norm is bounded by 1 and so almost sure
convergence of the optimal filter implies
E πY1:n π˜Y1:n p 0, n ,
|| n − n ||var → →∞
by dominated convergence.
(cid:0) (cid:1)
Moreover,for some values of p we can specify the rates of this convergence:
Theorem 9. For p 0, u0 lnE(Wβ) we find
∈ −B¯ 1
(cid:16) (cid:17)
E πY1:n π˜Y1:n p = (ρn), n ,
|| n − n ||var O →∞
where ρ= E(W1β)ep(cid:0)uB0¯ p+pβ and B¯ = o0(cid:1)−2u0 β(β+1)− 2βo20u0 + β(βu+201)o20 21.
(cid:16) (cid:17) (cid:16) (cid:17)
10 BIRGITDEBRABANTANDWILHELMSTANNAT
Note that the theorem achieves lower rates than those in the pure Gamma case
since
p
E(W1β)epuB0¯ p+β ≥ E(W1β)p+pβ ≥ E(W1pp+ββ) ≥ E(W1p).
The proo(cid:16)f is based o(cid:17)n Theorem 5 and the bound (14) for the total variation
distance. ConsiderfirstthefollowingLemmaspecifyingthebehavioroftherandom
index J introduced in the proof of Thm. 5.
δ
Lemma 10. Let 0<δ <E(W1β)−β1. For
J = inf k N:bjY >δj for all j k
δ j
{ ∈ ≥ }
we find
E Xβ
0
(15) P(J >n) qn, n N ,
δ ≤ Γ(β+(cid:16)1)(1(cid:17) q) ∈ 0
−
where q =δβE(Wβ).
1
Proof. Using (9) we have for n N
0
∈
∞
P(J >n) = P bjY δj P bjY δj
δ ∪∞j=n{ j ≤ } ≤ j ≤
j=n
(cid:0) (cid:1) X (cid:0) (cid:1)
E X0β qn
.
≤ Γ(β(cid:16)+1(cid:17)) · 1 q
−
(cid:3)
Proof. of Thm. 9: First note that for s 0
≥
∞
E esJδ =1+(es 1) esnP(Jδ >n).
−
n=0
(cid:0) (cid:1) X
Due to Lemma 10 the expectation is finite if esq <1, where q =δβE(Wβ).
1
1
Now let δ = E(W1β)epuB0¯ −p+β. Then δ satisfies 1 < δ < E(W1β)−β1. Now,
Lemma 10 applie(cid:16)s and we obt(cid:17)ain from (14) that
E πY1:n π˜Y1:n p
|| n − n ||var
(cid:0)≤ Q||h0||LipE ep(cid:1)B¯δ−δ1eδpuBp0¯nJδ1l{Jδ≤n}+epuB0¯n1l{Jδ>n}!
≤ Q||h0||LipepB¯δ−δ1E(cid:16)eδppunB0¯Jδ(cid:17) +epuB0¯nP(Jδ >n)
 
≤ Q||h0||LipepB¯δ−δ1E(cid:16)eδppunB0¯Jδ(cid:17) + ΓE(β(cid:16)X+0β1(cid:17))epuB0¯nqn,
 
whichtends to 0 as (ρn) since epuB0¯q =δ−p =ρ andρ<1 due to the assumptions
on p. O (cid:3)