We call a connected graph T a tree if it is an infinite and locally finite, with a conspicuous node o called the root and without loops or cycles. In this work, we restrict the degrees of the nodes to a number of no less than 2. Let σ, τ be nodes of T. Write \(\tau< \sigma\) if τ is on the unique path connecting o to σ, and \(\vert\sigma\vert\) for the number of edges on this path. For any two nodes σ, τ, denote by \(\sigma\wedge\tau\) the node farthest from o satisfying \(\sigma\wedge\tau < \sigma\), \(\sigma\wedge\tau< \tau\).
Some useful notations are listed as follows: \(|A|\) is the number of elements in the set A, \(L_{n}\) is the nodes in level n of tree T, and \(L_{0}\) is the set of the roots o, \(T^{(n)}\) is the nodes in level 0 to n of tree T, \(T^{(n)}\setminus\{o\}= \{ T^{(n)} \mbox{ excluding the root } o \}\), \({1}_{t}\) is the first predecessor of t, \({2}_{t}\) is the second predecessor of t. See Figure 1 for an example.
The study of tree-indexed processes began at the end of 20th century. Since Benjamini and Peres [1] introduced the notion of the tree-indexed Markov chains in 1994, much literature (see [2–9]) studied some strong limit properties for Markov chains indexed by an infinite tree with uniformly bounded degree. Meanwhile, there are many authors (see [10–12]) who tried to give the limit properties of Markov chains indexed by a class of non-uniformly bounded-degree trees.
This work, motivated by Peng (2014), mainly considers a kind of non-uniformly bounded-degree trees and studies some strong limit properties, including the strong law of large numbers and AEP with a.e. convergence, for nonhomogeneous Markov chains indexed by a controlled tree, which permits some of the nodes to have an asymptotic infinite degree. The outcomes can generalize some well-known results. The technical route used in this paper is similar to that in [13], some of the related notations in this paper are the same as [13].
Definition 1
(see [14])
Let T be a tree, S be a states space, \(\{X_{\sigma},\sigma\in T\}\) be a collection of S-valued random variables defined on the probability space \((\Omega,\mathcal{F},P)\). Let
$$ p=\bigl\{ p(x),x\in S\bigr\} $$
(1)
be a distribution on S, and
$$ \bigl(P_{t}(y|x)\bigr), \quad {x,y\in S}, t\in T, $$
(2)
be stochastic matrices on \(S^{2}\). If, for any vertices \(t\in T\),
$$\begin{aligned}& \mathrm{P}(X_{t}=y|X_{1_{t} }=x \mbox{ and } X_{s} \mbox{ for } t \wedge s < 1_{t}) \\& \quad =\mathrm{P}(X_{t}=y|X_{1_{t} }=x)=P_{t}(y|x), \quad x, y\in S \end{aligned}$$
(3)
and
$$\mathrm{P}(X_{o}=x)=p(x),\quad x\in S, $$
then \(\{X_{t},t\in T\}\) will be called S-value nonhomogeneous Markov chains indexed by a tree with the initial distribution (1) and transition matrix (2), or they will be called tree-indexed nonhomogeneous Markov chains. If the transition matrices \((P_{t}(y|x))\) have nothing to do with t, i.e., for all \(t\in T\),
$$\bigl(P_{t}(y|x)\bigr)=\bigl(P(y|x)\bigr), \quad {x,y\in S}, $$
\(\{X_{t},t\in T\}\) will be called S-value homogeneous Markov chains indexed by tree T.
We set an integer \(N \geq0\), \(d^{0}(t):=1\), and denote by
$$ d^{N}(t):=\bigl\vert \{{\sigma\in T:{N}_{\sigma}=t}\}\bigr\vert , \quad N\geq1, $$
(4)
the number of t’s Nth descendants. We assume that, for any integer \(N\geq0\), there are constants \(\delta >0\), and positive integers \(M_{k}\), \(k=0, 1, 2,\ldots \) , such that
$$ \frac{|\{t\in T^{(n)}: d^{N}(t)>M_{N} \} | }{|T^{(n+N)}|} \leq\frac {1}{(1+\delta)^{ d^{N}_{n}}} $$
(5)
uniformly holds for all \(n\geq0\), where \(d^{N}_{n}= {\max_{t\in T^{(n)} }}\{d^{N}(t)\}\).
Definition 2
We call T a controlled tree if it is a non-uniformly bounded-degree tree when the assumption (5) holds.
From the assumption (5) we can find that some of the nodes on a controlled tree may have an asymptotic infinite degree. The following three remarks indicate that controlled tree models include some well-known models such as Cayley trees (of course homogeneous trees) and uniformly bounded-degree trees.
Remark 1
A Cayley tree \(T_{C,m}\), of which each vertex has m descendants, satisfies the above condition (5). Actually, in such a tree, \(d^{N}_{n}=m^{N}\), hence \(|\{t\in T^{(n)}: d^{N}(t)> m^{N}\}|=0\).
Remark 2
If we consider any uniformly bounded-degree tree, then there are some \(a>0\) such that \(d^{N}_{n}\leq a^{N}\), \(|\{t\in T^{(n)}: d^{N}(t)> a^{N} \}|=0\), which indicates that uniformly bounded-degree trees conform to the assumption (5).
Remark 3
In this paper, the condition (5) can imply the case in Peng [13]. The assumption (5) in [13], which we denote by (5a), is
$$ \max\bigl\{ {d^{N}(t)}:{t\in{T^{(n)}}}\bigr\} \leq O\biggl(\ln{ \frac {|T^{(n+N)}|}{|T^{(n)}|}}\biggr), $$
(5a)
where
$$O(n)=\biggl\{ c_{n}: 0< \limsup_{n\rightarrow\infty} \frac{ c_{n}}{n}\leq c, c \mbox{ is a constant}\biggr\} . $$
In fact, (5) is equivalent with
$$ d^{N}_{n} \leq\log_{1+\delta} \frac{|T^{(n+N)}|}{|\{t\in T^{(n)}: d^{N}(t)>M_{N} \} | }. $$
(6)
Meanwhile, (5a) is equivalent with
$$ { d^{N}_{n}}\leq O\biggl(\ln{\frac{|T^{(n+N)}|}{|T^{(n)}|}} \biggr). $$
(7)
Obviously,
$$ \bigl\vert \bigl\{ t\in T^{(n)}: d^{N}(t)>M_{N} \bigr\} \bigr\vert \leq\bigl\vert T^{(n)}\bigr\vert . $$
(8)
Then, for all \(\delta>0\), combining (6), (7), and (8), we arrive at
$$ O\biggl(\ln{\frac{|T^{(n+N)}|}{|T^{(n)}|}}\biggr) \leq\log_{1+\delta} \frac {|T^{(n+N)}|}{|\{t\in T^{(n)}: d^{N}(t)>M_{N} \} | }. $$
(9)
Hence, by (9) we know the trees with (5a) holding are special cases of the controlled tree model.
The above three remarks indicate that the tree models introduced in this work are extensions of [7, 15] and [13]. Without additional statement, the trees referred to in the following are all infinite, local finite trees with assumption (5) holding.
Now we give some useful notations. Let \(\delta_{k}(\cdot)\) be the indicator function, i.e.,
$$\delta_{k}(x)= \left \{ \textstyle\begin{array}{l@{\quad}l} 1, & \mbox{if }k=x, \\ 0, & \mbox{or else}. \end{array}\displaystyle \right . $$
For given natural integer \(N\geq0\), write
$$ S_{k}^{N}\bigl(T^{(n)}\bigr):=\sum _{t\in T^{(n-N)}}\delta_{k}(X_{t})d^{N}(t). $$
(10)
By (10), we have
$$ \sum_{k\in S}S_{n}^{N} (k)=\bigl\vert T^{(n)}\bigr\vert -\bigl\vert T^{(N)}\bigr\vert . $$
(11)
Denote
$$ H_{n}(\omega)=\sum_{t\in T^{(n)}\setminus\{o\} }g_{t}(X_{{1}_{t}},X_{t}) $$
(12)
and
$$ G_{n}(\omega)=\sum_{t\in T^{(n)}\setminus\{o\} }E \bigl[g_{t}(X_{{1}_{t}},X_{t})|X_{{1}_{t}}\bigr]. $$
(13)