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GENERALIZATIONS OF SHANNON-MCMILLAN THEOREM
S HU -T EH C HEN M OY
Vol. 11, No. 2
December 1961
GENERALIZATIONS OF SHANNON-MCMILLAN THEOREM
SHU-TEH C. MOY
1«, Introduction, Let X b e a non-empty set and ^ be a <7-algebra of
subsets of X. Consider the infinite product space Q — n»=-~ Xn where Xn — X
for n = 0, ± 1 , ±2, • • • and the infinite product tf-algebra J^~ = II~=-oo ^
where ^—S/1
for w = 0, ± 1 , ± 2 , • • • . Elements of -0 are bilateral
infinite sequences {• • •, X-19 x09 x19 • • •} with xn £ X. Let us denote the
elements of Q by o). If <w = {•••, #_!, #0, #2, • • •} xw is called the nth
coordinate of co and shall be considered as a function on Q to X. Let
T be the shift transformation on Q to 42: the nth coordinate of Too is
equal to the n + lth coordinate of co. For any function g on Q, Tg is
the function defined by Tg(a)) = 0(Ta>) so that T&w = jcn+1. We shall
consider two probability measures ft, v defined on JC Let Qn = n*=i X*
where X4 = X, i = 1, 2, • • •, n and ^ = n?=i ^ f where ^f=s^i
=
1,2,---,n.
Then 42X = Z and ^ f = ^ Let ^ , n , m ^ w , w = 0, ± 1 , ± 2 , •••,
be the (7-algebra of subsets of Q consisting of sets of the form
[a) = {•••, #_!, o?o, o;3, • • • } : (xm, xm+1, •••, x j e E]
where E e ^ L w 4 1 . Let ^oo lW be the a-algebra generated by Um=-i ^ . » Let j«m>w, vm<n be the contractions of p, v, respectively, to J^,n and P-^^,
v-co,n be the contractions of p, v, respectively, to J^i^^.
Throughout
this paper vm>n is assumed to be absolutely continuous with respect to
Pm,n, vm,n < Pm,n, for m < n, n = 0, ± 1 , ±2, • • •. Let/ m ,« be the derivative of vm>n with respect to ftm,n9fm,n = dvmjdpm>n.
fmM is ^ n measurable
and nonnegative. / m w is also positive with v probability one. Hence
Vfm.n is well defined with v probability one. A fundamental theorem
of Information Theory by Shannon and McMillan may be considered as
a theorem concerning the asymptotic properties of fm>n as n—* oo. The
theorem may be stated as follows: Let X be a finite set of K points
and Sf7 be the a-algebra of all subsets of X. Let v be any stationary
(T invariant) probability measure on j^~~ and p. be the equally distributed
independent (product) measure. Then wUog/i,,, converges in L^v). In
particular, if v is ergodic, the limit function is equal to log K — H with
v probability one where iTis the entropy of v measure [3] [8]. Generalizations to arbitrary X, c/7 were first studied by A. Perez. He introduced
an A,,, condition on v as follows, v is said to satisfy A^ condition if
v_oo,» is absolutely continuous with respect to v_OOtQ,p1>n for n = 1, 2, • • •.
He proved the following theorem. If vf p are stationary and p is the
product (independent) measure on ^
and if
Received May 31, 1960. This work was supported in part by a summer research grant
from Wayne State University.
705
706
SHU-TEH C. MOY
(a) limn_>oo n"M log f1>ndv exists and is finite,
(b) v satisfies condition AM
then {n~l\ogf1>n} converges in L^v) [6]. Later Perez announced that
the theorem remains to be true for any stationary measures fi, v [8].
The present writer proved that for Markovian fi, v with v being stationary
and fi having stationary transition probabilities the v-integrability of
log/i,2 implies the Lx{v) convergence of {nr1 log fun}. The proof is based
on an iteration formula for f1>n [4]. In this paper we shall study the
case that v is stationary and fi is Markovian with stationary transition
probabilities. It shall be proved that the condition
(c) j (log/1)W - log f^-Jdv g M < co for n = 1, 2, 3, • • • implies the
L^v) convergence of {wMog/jJ. In fact the conditions (c) and (a) are
equivalent for this case, so that the theorem is a generalization of the
theorem of Perez given in [6]. The proof is conducted along similar
lines used by McMillan. The crucial step is proving the L^v) convergence
of {log/_w0 — log/-w,-i}. The condition (c) is shown to be necessary and
sufficient for this convegence.
2» Generalizations of Shannon-McMillan theorem* Let x, £? Q,
J^i2nf^nf^.n,t*m.nfVm.n?fm.n
be as in /. Notations for conditional
probabilities and conditional expectations relative to one or several random
variables will be as in [2], Chapter 1, § 7. A probability measure on
^ is Markovian if, for any A e t 5 f m < w w = 0, ± 1 , ±2, • • •
P[xn e A I xm, • • •, xn^] = P[xn 6 A I a?n_J
with probability one. A Markovian measure is said to have
transition
probabilities if for any A e y and any integer n
stationary
P[xn e A I Xn-J = TnP[x0 e A \ a.J
with probability one. In this paper, since we have two probability
measures /^, v, we need to use subscripts ft, v to indicate conditional probabilities and conditional expectations taken under ft, v respectively. For
any E c Q,IE, the indicator of E, is the real valued function on Q
defined by
IE(o)) =z\ if a) e E
= 0 if co $ E .
The log in this paper is the logarithm with base 2.
LEMMA 1.
(1)
Define
»
v'm,n on J*£ffI by
GENERALIZATIONS OF SHANNON-MCMILLAN THEOREM
707
then v'm>n is a probability measure on ^ > w with v'Wtn(E) — vmin(E) for
E e J ^ - x . Furthermore vm<n < v'm<n with
Proof.
I ^ . n - ! I Xm, • • • , X
= \ fm.n-idp .
JE
Hence v'mM is a probability measure on ^ , w . Furthermore, for E e ^ , ,
fm.ndfi - [ (fm.Jfn.n-l)fm.n-ldf*
JE
JE
Hence vm>n is absolutely continuous with respect to v'm>n and dvm>nldv'mtn —
Jm.nlJm.n—i*
THEOREM 1. If v is stationary and fx is Markovian with stationary
transition probabilities then
\ ** )
Jm,nlJm,n—i
•*•
\Jm—n,oUm—n,—i)
with v probability one for all m < n, n — 0, ± 1 , ± 2 ,
Proof. If fi is Markovian and has stationary transition probabilities
then for any A e Sf
Py\.®n ^ A I Xm, • • •, #w-iJ = r\\Xn
6 A
%n-\\
n
- T P»[x0 e A I aj.J
with ix probability one and, therefore, also with v probability one. Hence
for any Ae^fBe
^T_m
v'm,n[xn e A, (xm, . ••,&„_!) e 5 ]
=
\
Pt&n
6 A I flJTO, •
708
SHU-TEH C. MOY
TnP,Hx0 e A
= [
J[(xm_1,---,a;_1)6£]
xQ e A I x ^
It follows that
*4,»[(a«, • • •, o?n) e C]i= v'm-n>o[(xm-n, • •., a;0) e C]
for every C e ^ _ m + 1 .
^^m.nl^^m.n
Since by Lemma 1
~ Jm,nlJm,n-l> ^^m-n.ol^^m-n.O
==
Jm—n,olJm-n,—l
(2) follows easily.
LEMMA 2. 7/ /^ is Markovain
extension of i42,0 ^
Proo/.
and m1 < m2 < 0 t/ieti ^ l t 0 is an
For any A e £? /3 e ^T m 2
v'mv0[x0 e A , (ajTOa, • • •, tf-x) e B ]
*[%o e A \ x
m
,
---tX^
= \
= ^m2lo[^o e A,(a?TO2, • • -, a?_i) e B] .
It follows that
for every E e
THEOREM
2
2. If fi is Markovian and m1 < m2 < 0
(3)
Proof. By Lemma 2 i4 li0 is an extension of v«it0 to ^ 1 § 0 . Since
^ . o < ^ . o , ^ 2 ( 0 < v;2,0 by Lemma 1, dvmjdv'm2>, is the conditional expectation of dvmitOldv'mvO relative to ^ 2 , 0 under the measure v'mim0. Jensen's
GENERALIZATIONS OF SHANNON-MCMILLAN THEOREM
709
inequality for conditional expectation implies that
0
Hence
(4)
0 g J log (dvmjdv'mjdv
^ j log
(dvmJdvmv0)dv
and (3) follows from (4) and Lemma 1.
THEOREM 3. / / [i is Markovian then {log/wl 0 — log/TO,-i} converges
with v probability one as m —> — oo. The limit function may take ± GO
as its values.
Proof. It is sufficient to prove that {/m,-i//wM} converges with v
probability one as m —> — oo. Since vmi0 is absolutely continuous with
respect to v'm>0 and dvmiQldv'm>0 = fmjfm,-i
by Lemma 1, fm,-Jfm.o
is t h e
derivative of vmi0 continuous p a r t of v'm>0 with respect to vm>Q. Since, by
Lemma 2, ^ 1>O is an extension of i4 lp0 if mx < m 2 , {—/-fc)-i//-fc>0, ^*.o>
fc ^ 1} is a v semimartingale ([2] p p . 632). Since
the semimartingale convergence theorem implies that {f-k-Jf-kiQ} converges
with v probability one as k —> oo.
The following lemma may be considered as an improvement of a
theorem by A. Perez ([6] Theorem 7; pp. 194).
LEMMA 3. Let /31 c j32 c • • • 6e a sequence of a-algebras of subsets
of Q and /3 be the a-algebra generated by \Jk /3k. Let </>, X be two probability measures defined on ft and 0ft, Xfc be the contractions of </>, X,
respectively, to ftk. If $k is absolutely continuous with respect to Xk
for k = 1, 2, • • • and if there is a finite number M such that
\ log {d$
for
k — 1, 2, • •
( i ) cj> is absolutely continuous
(ii) log (d<PldX) is <fi integrable
^ M
with respect to X,
and there exists
Km I log (d<t>kldxk)d4> = \ log {d<t>\dX)d<f> ,
710
SHU-TEH C. MOY
(iii)
{log (d4>kld\k)} converges
in Lx(4>) to log (d(f>ldX).
Proof.
( i ) Let hk — d</)JdXk. Then {hk, /?*., k ^ 1} is a martingale under
X measure. Now
M ^ j log (d<t>kldXk)d<t> = [ (log hk)hkdX .
and
( 5)
M + i ^ [(/^ log ft* + J ) d \ ^ (log ^) (
ftfcdA,
}
Hence
(
KdX^ilognyW+i)
J(hk^n)
so that I
hkdX —> 0 as % —• oo, uniformly in k. Hence {hk} converges
with X probability one and also in LX(X) ([2] Theorem 4.1, pp. 319). Let
the limit function be h. Then I hdX = 4>(A) for all A e \Jkftk and so
for all A e /3. This proves that <£ is absolutely continuous and that
h = (d^/dX).
(ii) The sequence {/^ log Z^} converges with A, probability one to
h log /&. Since the functions hk log /^ are bounded below uniformly by
the number J,
\h log MX ^ lim lfcfc log /ifcd\ — lim^l log hkd<t> ^ M .
Hence hlogh is X, integrable. Since the real valued function | l o g | is
continuous and convex, h± log /^, h2 log /t2, • • •, h log ft constitute a semimartingale under the measure X([2], Theorem 1.1, pp. 295). Hence
\h± log hxdX ^ \ft2 log ft2dX ^ • • • ^ ft logftdA,,
so that lim^oolftfclogftfcdX exists and is equal to \ftlogftdX. Now
I I log h\d<t> — \ft I log ft I dX — I IftlogftI dX, ,
hence log ft is ^ integrable and
( 6)
(logftd<£= Ift logftcfX,= lim \hk log ftfc dX = lim \ log ftfc d0 .
J
1
J
k->°o J
Inequality (5) was pointed out by the referee.
shortened by following his suggestions.
fc-*oo
J
The proof of Lemma 3 was much
GENERALIZATIONS OF SHANNON-MCMILLAN THEOREM
711
(iii) Since hx log huh2\ogh2T • • •, h log h constitute a semimartingale
under the measure X, we have, for E e /3kf
l hjcloghjcdX g \ hk+1 loghk+1dX g \
JE
JE
hloghdX.
JE
Hence
\ loghkd(f)^
\ loghjc^dcj) ^ \ \oghdcj> ,
JE
JE
JE
so that log 7^, \ogh2, • • •, logh constitute a semimartingale under the
measure 0. Hence (ii) implies that log hk are uniformly <f> integrable and
{loghjc} converges to log h in L^) ([2], Theorem 4.1s, pp. 324).
THEOREM
4. If ft is Markvian
and there is a finite number M
such that
^,-^dv rg M
for
m— — 1, —2, ••• then {log/m>0 — \ogfm<_^\ converges in Lx(v) as
m —* — oo.
Proof.
By Lemma 2 v'mv0 is an extension of vrm^ if m1 < m2 < 0
and
If there is a probability measure i/ defined on the tf-algebra generated
by Um=-i^m,o which is an extension of v'm>0 for m = — 1 , —2, •••, then
the conclusion of the theorem follows easily from Lemma 3. If X is
the real line and if S^ is the tf-algebra of Borel sets then the existence
of i/ follows from the Consistency Theorem of Kolmogorov. For the
general case we shall proceed by using the usual representation by space
Qf of sequences of real numbers as follows:
Let
9lC — / ~JC,0lJ -16,-1 *
L e t G b e t h e m a p of Q into t h e space Qf of real sequences { £ , | 2 , • • • }
defined b y
G((o) =
Considering £k as functions on Qr we have
-
gk(co) .
Let /3k be the collection of Borel subsets of Q' which are determined
by conditions on £lf | 2 , • • •, gfc and ft be the collection of all Borel subsets
712
SHU-TEH C. MOY
of Q'. Let 0 be the probability measure on /3 and </>fc, Xk be the probability measures on j3k defined by
Xk(E) = v'-UG'lE)
.
{gk} converges in L^v) if and only if {£&} converges in L^).
Now Xk
are consistent; Kolmogorov's Consistency Theorem implies the existence
of a probability measure X on ft which is an extension of every Xk and
d(f>kldXk — ffc. Hence Lemma 3 is applicable and the Z/i(</>) convergence
of {£,.} is obtained.
THEOREM
ary transition
5. If v is stationary
probabilities and if
and pt is Markovian
with
station-
and if there is a finite number M such that
for n — 1,2, ••• then n~l\ogfQ%n converges in Lx(v) as n-+oo.
In
particular, if v is ergodic, the limit is equal to a nonnegative constant
with v probability one.
Proof. By Theorem 4 {log/ m0 — log/^.-J converges in Lx(v) as
Let h be the Lx{v) limit of the sequence. Let h be the
m_>_oo.
Lx{v) limit of the sequence {n'1 £?=i T'h}. By Theorem 1 /0>B//0,n-i =
r n (/- B . 0 //- w ,-i), hence
-1 logr/o.. = n-1 log/ 0 , 0 + w1 £ 2" log
(f-Jf-t.-i)
^ W-1 t j I T* log (f-i.Jf-i.-r) ~ T'h I dv
^~x S r*^ - h I dv -> 0 as n -> oo .
Thus the Lx(v) convergence of {^"Mog/o.J is proved.
The limit is h
GENERALIZATIONS OF SHANNON-MCMILLAN THEOREM
713
which is the Lr{v) limit of {n'1 2?=i T*h}. If v is ergodic
h — \hdv
with v probability one and
\hdv - lim [[log/w,o - log fn.-Jdv
^ 0.
COROLLARY 1. Under the hypothesis of Theorem 5 if v is
and ergodic but not Markovian then v is singular to pt.
stationary
Proof. If [X is Markovian but v is not Markovian then there is a
positive integer n0 such that
i"[/o.» o - 1 =3 fc /o.n o ]>O.
For, if for every positive integer n
fAAn-i * An] = 0
then
jry\Xn 6 J± XQ, • • •, # w _ij — ±y\Xn 6 A. \ ^ w -iJ
w i t h v probability one for every A e y
N o w since
Jo,no-i
=
a n d v is M a r k o v i a n i n s t e a d .
-^^L/o.Wo I ^o> * * * 9 %no-i\
and the function I;log! is strictly convex, hence
5/o.»0los/o,«0d/i - JZo.no-! log/o,»o-idj" > 0
so that
5[logZo,,0 - logZo.»0-i]d^ > 0 .
Since \[logZo,» — logZo,w-i]^^ is non-decreasing in n,
lim l[logZo,» - logZo>w-i]^^ = a> 0 .
Now v is ergodic; the Lx(v) limit h of {^"MogZo.J is equal to a with y
probability one. Let nlf n2, • • • be a sequence of positive integers for
which {wjb"1 log Zen*.} converges with v probability one to a so that {1/Zo.nJ
converges to 0 as nk—> oo. Let ^ ' be the ^-algebra generated by
\Jn^o,n and let fi^,, [i^, be the contractions of [i, v, respectively, to _^ r '.
Since 1/Zo,n is the derivative of v-continuous part of fJt0i7l with respect
714
SHU-TEH C. MOY
to vOin, {l//o.»} converges with v probability one to the derivative of v-continuous part of fi' with respect to 1/ by a theorem of Anderson and
Jessen [1], Now we have
lim l// 1>n - 0
with v probability one and fi is singular to i/. Hence fi, v are singular
to each other.
Extensions of Theorem 5 and Corollary 1 to if-Markovian fi are
immediate.
3. Discussion* As was mentioned in the introduction the crucial
step in establishing Theorem 5 is to prove the L2(v) convergence of
{log/_W(0 — log/- w ,-i}. If fi is the product (independent) measure on
^
the measure vf in the proof of Theorem 4 is actually v_oo-i x fiQ>Q. Thus
condition (c) or, equivalently, condition (a) implies condition (b) in the
introduction. In [7] it is stated that the condition (b) is necessary for
the LXv) convergence of {log/_„,„ —log/_„,_!} ([7] Theorem 2(b)). A
simple is as follows. Let X be the real line and £f be the collection
of all Borel sets. Let v — fi and distribution of x0 be Gaussian. Let
v(x0 = xx) = fi(x0 = xx) = 1. Then v_li0 is singular to v_1(_! x v0>0, however the Lx(v) convergence of {log/_ wo — logZ-^.J is trivially true since
f
= 1
REFERENCES
1. Erik Sparre Anderson and Borge Jessen, Some limit theorems in an abstract set, Danske
Vic. Selsk. Nat.-ys. Medd. 22, No. 14 (1946).
2. J. L. Doob, Stochastic Processes, John Wiley and Sons, Inc., New York.
3. B. McMillan, The basic theorems of information theory, Annals of Math. Statistics, 2 4
(1953), 196-219.
4. Shu-Teh C. Moy, Asymptotic properties of derivatives of stationary measures, Pacific J.
Math., 1O (1960), 1371-1383.
5. A. Perez, Sur la theorie de V in formation dans le cas d'un alphabet abstrait, Transactions
of the First Prague Conference on information theory, statistical decision functions, random
processes, Publishing House of the Czechoslavak Academy of Sciences, Prague, (1957) 183208.
6.
, Sur la convergence des incertitudes, entropies et informations echantillon
(sample) vers leurs vraies, Transactions of the First Prague Conference on information
theory, statistical decisions functions, random processes, Publishing House of Czechoslavak
Academy of Sciences, Prague (1957), 209-243.
7
, Information theory with an abstract alphabet, generalized forms of McMillan's
limit theorem for the case of discrete and continuous times, Theory of Probability and its
Applications, (An English translation of the Soviet Journal Teoriya Veryoatnostei i ee Primeneniya) Vol. 4, No. 1 (1959), 99-102.
8. C. E. Shannon, The mathematical theory of communication, Bell System Technical
Journal, 2 7 (1948), 379-423, 623-656.
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Pacific Journal of Mathematics
Vol. 11, No. 2
December, 1961
Tsuyoshi Andô, Convergent sequences of finitely additive measures . . . . . . . .
Richard Arens, The analytic-functional calculus in commutative topological
algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Michel L. Balinski, On the graph structure of convex polyhedra in
n-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
R. H. Bing, Tame Cantor sets in E 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cecil Edmund Burgess, Collections and sequences of continua in the plane.
II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
J. H. Case, Another 1-dimensional homogeneous continuum which contains
an arc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lester Eli Dubins, On plane curves with curvature . . . . . . . . . . . . . . . . . . . . . . .
A. M. Duguid, Feasible flows and possible connections . . . . . . . . . . . . . . . . . . .
Lincoln Kearney Durst, Exceptional real Lucas sequences . . . . . . . . . . . . . . . .
Gertrude I. Heller, On certain non-linear opeartors and partial differential
equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Calvin Virgil Holmes, Automorphisms of monomial groups . . . . . . . . . . . . . . .
Wu-Chung Hsiang and Wu-Yi Hsiang, Those abelian groups characterized
by their completely decomposable subgroups of finite rank . . . . . . . . . . .
Bert Hubbard, Bounds for eigenvalues of the free and fixed membrane by
finite difference methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
D. H. Hyers, Transformations with bounded mth differences . . . . . . . . . . . . . . .
Richard Eugene Isaac, Some generalizations of Doeblin’s
decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
John Rolfe Isbell, Uniform neighborhood retracts . . . . . . . . . . . . . . . . . . . . . . . .
Jack Carl Kiefer, On large deviations of the empiric D. F. of vector chance
variables and a law of the iterated logarithm . . . . . . . . . . . . . . . . . . . . . . . .
Marvin Isadore Knopp, Construction of a class of modular functions and
forms. II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gunter Lumer and R. S. Phillips, Dissipative operators in a Banach
space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nathaniel F. G. Martin, Lebesgue density as a set function . . . . . . . . . . . . . . . .
Shu-Teh Chen Moy, Generalizations of Shannon-McMillan theorem . . . . . . .
Lucien W. Neustadt, The moment problem and weak convergence in L 2 . . . .
Kenneth Allen Ross, The structure of certain measure algebras . . . . . . . . . . . .
James F. Smith and P. P. Saworotnow, On some classes of scalar-product
algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dale E. Varberg, On equivalence of Gaussian measures . . . . . . . . . . . . . . . . . . .
Avrum Israel Weinzweig, The fundamental group of a union of spaces . . . . .
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