Astrong limittheoremof the entropy density of nonnegative continuous Markov information source on a non-homogeneous tree
Jin Shaohua＊，1, Wan Yanping, Sun Shuguang
Hebei University of Technology, Tianjin，300401, China
Abstract
In recent years, tree indexedstochastic process has become one of the research
directions for studying in the probability theory. The strong limit theorems has
been one of the central issues of the international probability theory .In this paper, a
strong limit theorem of the entropy density of nonnegative continuous Markov information source on a non-homogeneous tree is given.
Key words: non-homogeneous tree, martingale, Markov information source,entropy density，strong limit theorem
MSC(2010) 60F15
1. Induction
In recent years, tree indexedstochastic process has become one of the research
directions for studying in the probability theory. The strong limit theorems hasbeen one of the central issues of the international probability theory . (c.f. [1] and [2]). The limit properties of relativeentropy density is an important issue in information theory. In this paper, through constructing a non-negative martingale and applies Doob's martingale convergence theorem to the research of a.e. convergence, astrong limit theorem of the entropy density of nonnegative continuous Markov information source on a non-homogeneous tree is given.
Let be an infinite tree whose root vertex is , is a positive integer set. If every vertex on level has adjacent vertices , we call the tree
is a Bethe tree in broad sense or a Cayley tree in broad sense.Particularly, to
* Corresponding author.
E-mail address:
1 Supported by the education department of Hebei province of China(ZD2014051).
classify the non-negative integer set ,we divide into several sets as follow:
When , let ,,( is a positive integer and not all of them is 1). In this way , we get a particular tree. It is convenient for us to denote by . We denote be the set of all vertices on level , be the subtree of containing the vertices from level to level . We denote be the number of the vertices of the subtree , , and be the sons of the vertex .
Definition 1 Let be a probability space ，be a information source defined on and taking values in continuous state.The joint density function of is
（1）
Let （2）
is called relative entropy density of .
Let
， （3）
be the initial distribution of ，and the following (4)
， （4）
be a regular conditional probability family. If
（5）
we call a transition density function. In this paper, is denoted by ,that is
（6）
Ifbe a sequence of transition density functions of , and be the probability density function of the corresponding initial distribution of , then is called a continuous non-homogeneous Markov information source with initial distribution （3）and regular conditional probability family （4）.Hence the joint density function of is
（7）
The relative entropy density of is
(8)
Definition 2Let be a information source with distribution（1）. Let
and （9）
and are called relative entropy density deviationand sample relative entropy rateof relative to the non-homogeneous Markov information source with the joint density function（7）respectively.
Definition 3 Let be a real number. Let
（10）
is calledLaplace transformof function.
2. Main results
Lemma 1 Let be a non-homogeneous Markov information source with joint density function（7）and be a information source with joint density function（1）. is defined by （10）. Let
（11）
(12)
(13)
Then is a non-negative martingale under measure .
Proof By（1），we have
（14）
By（11）,（12）and（13）, we have
（15）
where
（16）
Since
（17）
By（15）,（16）and（17）, we have
So is a non-negative martingale under measure .
Theorem 1 Let be a information source defined on and taking values in continuous state .The joint density function
of is given by （1）.and are defined as
before. If there is ，such thatis well-defined as . Let .If
（18）
then
(i)
（19）
(ii)
（20）
where
（21）
（22）
and
（23）
（24）
Proof By lemma 1 and martingale convergence theorem , ,such that
（25）
For ,by（9），（12），（13） and (25), we have
（26）
By (26) and the nature of superior limit ,we have
（27）
Lettingin（27）,we have
(28)
（i）Letting , dividing two sides of (27) by , we have
（29）
By the nature of superior limit and the inequality, we have
(30)
By （30）and the inequality,we have
(31)
Letting ，then reaches its maximum at , we have
（32）
Since ，by（32），we have
（33）
By（31）and（33），we have
， (34)
By（21），we have.As ，letting ，then we have
（35）
Hence ，so（23）holds.
Letting be the set of rational numbers in .,thenwe have.By （34），we have
， (36)
Since is continuous on, so by （21）and （28），for each ,there is , such that
（37）
By（36），we have
， （38）
By（37）and（38），we have
（39）
Since ，by（39），（19）holds.
(ii) Similar to the proof of (i). Omitted here.
Corollary 1 Let be a information source defined on and taking values in continuous state . The joint density function of is given by （1）. is defined as before. If（18）and hold，then
(40)
Proof Since, so the conditions of theorem 1 is satisfied，and we have
（41）
By（19）and（20），（40）holds.
References
[1] Weicai Peng,Weiguo Yang,Bei Wang. A class of small deviation theorems for
functionals of random fields on a homogeneous tree. Journal of Mathematical
Analysis and Applications.2010,361:293-301.
[2] Dong Y, Yang W G, Bai J F. The strong law of large numbers and the Shannon-McMillan theorem for nonhomogeneous Markov chains indexed by a Cayley tree. Statist Probab Lett, 2011, 81: 1883-1890.
* Corresponding author.
E-mail address:
1 Supported by the education department of Hebei province of China(ZD2014051).
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