S S e e m m i i g g r r o o u u p p s s a a n n d d P P r r o o b b a a b b i i l l i i t t y y : : F F r r o o m m r r e e p p r r e e s s e e n n t t a a t t i i o o n n t t h h e e o o r r e e m m s s t t o o P P o o i i s s s s o o n n a a p p p p r r o o x x i i m m a a t t i i o o n n D D i i e e t t m m a a r r P P f f e e i i f f e e r r I I n n s s t t i i t t u u t t f f ü ü r r M M a a t t h h e e m m a a t t i i k k C C a a r r l l v v o o n n O O s s s s i i e e t t z z k k y y U U n n i i v v e e r r s s i i t t y y , , O O l l d d e e n n b b u u r r g g C C o o n n f f e e r r e e n n c c e e i i n n h h o o n n o o r r o o f f P P a a u u l l D D e e h h e e u u v v e e l l s s P P a a r r i i s s , , J J u u n n e e 2 2 0 0 - - 2 2 1 1 , , 2 2 0 0 1 1 3 3
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Dietmar Pfeifer Institut für Mathematik Carl von … · Carl von Ossietzky University, Oldenburg Conference in honor of Paul Deheuvels Paris, June 20-21, 2013 . D. Pfeifer ...
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2 D. Pfeifer Conference in honor of Paul Deheuvels, Paris, June 20-21, 2013
IInnttrroodduuccttiioonn Important relationships between semigroups probability Feller semigroups Markov processes convolution semigroups Poisson approximation representation theorems Bochner and Pettis integral approximation theorems law of large numbers
3 D. Pfeifer Conference in honor of Paul Deheuvels, Paris, June 20-21, 2013
IInnttrroodduuccttiioonn Semigroups { }³ Í0 [ , ] of class ( )0C : on a Banach space ( )|T t t
(identity) = I(0)T
( ) for + =( ) ( )T s t T s T ³, 0s tt
-0
lim ( )t
T t f f = 0 Îf
w Î1,
for all
There exist constants such that ³M w£ tMe ³ 0t( )T t for Infinitesimal generator:
[ ]
= -0
1( ) lim ( )
t( )Af x T t
tI Î(ff x for )A and = =( ) ( ) ( )
dAT t f T t Af T t f
dt
4 D. Pfeifer Conference in honor of Paul Deheuvels, Paris, June 20-21, 2013
IInnttrroodduuccttiioonn Feller semigroups { }³( )| 0t { }³T t for standard Lévy processes | 0 :tX t
( )é ù+= ë ûtX f( ) ( )T t f x E f x Î = ( )UCB for
Infinitesimal generator:
[ ]( )
é ù+ -ë û( )tx X f xÎ( )f A= - =
0 0
1( ) lim ( ) ( ) lim
t t
E fAf x T t f x
t tI for
5 D. Pfeifer Conference in honor of Paul Deheuvels, Paris, June 20-21, 2013
IInnttrroodduuccttiioonn Feller semigroups { }³( )| 0t { }³T t for standard Lévy processes | 0 :tX t
( )é ù+= ë ûtX f( ) ( )T t f x E f x Î = ( )UCB for
Example: Brownian motion: m s= 2( ,t )t with tXP m sÎ >, 0 :
( ) sm
é ù+ -ë û= =0
( )( ) lim +
2
'( ) ''(2
t
t
E f x X f x)Af x f x f x
t
for { }Î = Î ( ) | ',f A f f Î''f
6 D. Pfeifer Conference in honor of Paul Deheuvels, Paris, June 20-21, 2013
IInnttrroodduuccttiioonn Feller semigroups { }³( )| 0t { }³T t for standard Lévy processes | 0 :tX t
( )é ù+= ë ûtX f( ) ( )T t f x E f x Î = ( )UCB for
Example: Brownian motion: m s= 2( ,t )t with tXP m sÎ >, 0 :
( )[ ] ( )
( )
+2 2
2
''( )f x t
ts
m
é ù+ -ë û é ù= + ë û
= + +2
( ) 1 1'( )
2
'( ) ''( )2
tt t
E f x X f xE X f x E X
t t t
f x f x
7 D. Pfeifer Conference in honor of Paul Deheuvels, Paris, June 20-21, 2013
IInnttrroodduuccttiioonn Feller semigroups { }³( )| 0t { }³T t for standard Lévy processes | 0 :tX t
( )é ù+= ë ûtX f( ) ( )T t f x E f x Î = ( )UCB
= Gt l> 0 :
for
Example: Gamma process: t with l( , )XP
( ) l¥
-
é ù+ -ë û= = ò00
( )( ) lim t u
t
E f x X f x f x + -( ) ( )u f xAf x e d
t uu
for { }Î = Î( ) |f D A f Î'f [pure jump process]
8 D. Pfeifer Conference in honor of Paul Deheuvels, Paris, June 20-21, 2013
IInnttrroodduuccttiioonn Feller semigroups { }³( )| 0t { }³T t for Lévy processes | 0 :tX t
( )é ù+= ë ûtX f( ) ( )T t f x E f x Î = ( )UCB
=t l> 0 :
for
Example: Poisson process: ) with l(XP t
( )[ ]l
é ù+ -ë û= = + -0
( )( ) lim ( 1) ( )t
t
E f x X f xAf x f x f x
t
for Î = ( )f A
9 D. Pfeifer Conference in honor of Paul Deheuvels, Paris, June 20-21, 2013
IInnttrroodduuccttiioonn Convolution semigroups { }³( )| 0tT t on with = p { }Î ¥1, :p
= f Îf
* for p( )t( )T t f
where ( ) 1),=p p p p( ) ( )(0), ( )(1), ( )(2t t t t Î is an infinitely divisible dis-
crete distribution, i.e.
+ = *( ) (s tp p( )s t ³, 0.s tp ) for
10 D. Pfeifer Conference in honor of Paul Deheuvels, Paris, June 20-21, 2013
IInnttrroodduuccttiioonn Convolution semigroups { }³( )| 0tT t on with = p { }Î ¥1, :p
= f Îf
* for p( )t( )T t f
(Example: Negative binomial convolution semigroup: )=p( ) ,t NB t
:p with
< <0 1p
æ ö+ - G +÷ç ÷= - =ç ÷ç ÷ G ⋅è øp
1 ( )( )( ) (1 )
( )t nt n
t n p pn
-(1 )!
t nt np p
t n
[ ][ ]
=
ì - ⋅ïïïï= - = í -ï - -ïïïîå0
1
ln(1
( ) lim ( ) ( ) (1 )( )
=
>
) (0) if 0
if 0kn
t
k
p f nAf n T t f n p
t f n k f nk
I Îf( )n
for
11 D. Pfeifer Conference in honor of Paul Deheuvels, Paris, June 20-21, 2013
IInnttrroodduuccttiioonn Convolution semigroups { }³( )| 0tT t on with = p { }Î ¥1, :p
= f Îf
lp( ) ( l> 0 :
* for p( )t( )T t f
Example: Poisson convolution semigroup: )t with = t
[ ][ ]) i
ll
ì - =ïï= - = íï - - >ïî0
(0) if 01( ) lim ( ) ( )
( 1) ( f 0t
f nAf n T t f n
f n f n ntI Îf for
12 D. Pfeifer Conference in honor of Paul Deheuvels, Paris, June 20-21, 2013
³( )
IInnttrroodduuccttiioonn Representation theorems for { }| 0t : T t
13 D. Pfeifer Conference in honor of Paul Deheuvels, Paris, June 20-21, 2013
IInnttrroodduuccttiioonn General case (A unbounded):
Hille’ s first exponential formula: with [ ]= -1
( )hh
I : hA T
( )
=0
( ) limexp hhT t f f ÎftA for
Hille-Yosida: with resolvent ll l
¥- -= - = ò1( ; ) ( ) ( ) ,tR A f A f e T t f dtI l w> :
0
( )ll¥
=( ) limexT t f f Îfp tB for
with [ ] for l l l l= -( ;B R I l w>)A
14 D. Pfeifer Conference in honor of Paul Deheuvels, Paris, June 20-21, 2013
IInnttrroodduuccttiioonn
General case (A unbounded): with [ ]= -1
( )hh
I : hA T
Kendall:
æ ö÷ç= + ÷ç ÷çè ø0( ) lim
n
h
tT t f fI Îf1/nA
n for
Shaw:
-
æ ö÷ç= -ççè ø0( ) lim
n
h
tT t f f
nI Îf÷÷1/nA for
[particular cases of Chernov’s product formula]
15 D. Pfeifer Conference in honor of Paul Deheuvels, Paris, June 20-21, 2013
PPrroobbaabbiilliissttiicc rreepprreesseennttaattiioonnss ooff ooppeerraattoorr sseemmiiggrroouuppss Some notation:
j é ù= =ë û å( ) :¥
=
=0
(N nN
n
t E t t P N )n
[probability generating function for a non-negative integer valued random variable N]
y¥
=
é ù é= = ùë û ë( ) : tX
X t E e ûå0 !
nn
n
tE X
n
[moment generating function for a non-negative real valued ran-dom variable X]
16 D. Pfeifer Conference in honor of Paul Deheuvels, Paris, June 20-21, 2013
z > 0Y
=
PPrroobbaabbiilliissttiicc rreepprreesseennttaattiioonnss ooff ooppeerraattoorr sseemmiiggrroouuppss Main Representation Theorem (Pfeifer 1984): Let N be a non-negative integer-valued random variable with and be a real-
valued random variable with such that
=( )E N
g( )E Y ( )j d <¥1N and
( )y d <¥2Y for some and Then for sufficiently large n, d1 > 0.>1 d2
jæ öé ùæ öç çê úç ççç ê úç è øè øë û
N E T ÷÷ ÷Î÷ ÷÷ ÷ [ ,
Yn
] with
wæ öj j y
é ù æ öæ ö æ ö÷ç ÷ç÷ ÷ç çê ú÷ £ ÷ç ÷ ÷çç ç÷ ÷÷ ÷ç çç ÷ç÷ê úç è ø è øè øè øë û,N N Y
27 D. Pfeifer Conference in honor of Paul Deheuvels, Paris, June 20-21, 2013
TThhee sseemmiiggrroouupp aapppprrooaacchh ttoo PPooiissssoonn aapppprrooxxiimmaattiioonn Theorem (Deheuvels & Pfeifer 1986): Under the assumptions of the startup framework, it holds
( ) ( ) ( ) { }( )l- 2max , ,( )v s t
(1) 2,k3v
l
rr
r d=
ì üï ïï ï= + - = + +í ýï ïï ïî þ 2 2
1
, 22
nS T A tA
ii
sP P p A e f e A A fI
if with p and =s=å
1
,n
kk
=t p=
=å1
n
k
s p=
=å1
n
kk
ld
-= .
ts
28 D. Pfeifer Conference in honor of Paul Deheuvels, Paris, June 20-21, 2013
45 D. Pfeifer Conference in honor of Paul Deheuvels, Paris, June 20-21, 2013
BBiibblliiooggrraapphhyy
[1] PFEIFER, D. (1984): Probabilistic representations of operator semigroups - a unifying ap-proach. Semigroup Forum 30, 17 - 34.
[2] PFEIFER, D. (1985): Approximation-theoretic aspects of probabilistic representations for op-erator semigroups. J. Approx. Theory 43, 271 - 296.
[3] DEHEUVELS, P. AND PFEIFER, D. (1984): A semigroup approach to Poisson approximation. Ann. Prob. 14, 663 - 676.
[4] DEHEUVELS, P. AND PFEIFER, D. (1986): Operator semigroups and Poisson convergence in se-lected metrics. Semigroup Forum 34, 203 - 224.
[5] DEHEUVELS, P. AND PFEIFER, D. (1987): Semigroups and Poisson approximation. In: New Per-spectives in Theoretical and Applied Statistics, M.L. Puri, J.P. Vilaplana and W.Wertz (Eds.), 439 - 448.
[6] DEHEUVELS, P. AND PFEIFER, D. (1988): On a relationship between Uspenky’s theorem and Pois-son approximations. Ann. Inst. Stat. Math. 40, 671 - 681.
[7] DEHEUVELS, P. AND PFEIFER, D. (1988): Poisson approximations of multinomial distributions and point processes. J. Multivar. Analysis 25, 65 - 89.
[8] DEHEUVELS, P., KARR, A., SERFLING, R. AND PFEIFER, D. (1988): Poisson approximations in selected metrics by coupling and semigroup methods with applications. J. Stat. Plann. Inference 20, 1 - 22.
[9] DEHEUVELS, P., PFEIFER, D. AND PURI, M.L. (1989): A new semigroup technique in Poisson ap-proximation. Semigroup Forum 38, 189 - 201.