June 27, 2025
The present study provides another look on Lamperti’s theorem on recurrence or transience of stochastic sequences. We establish a connection between Lamperti’s theorem and the recent result by the author in [V. M. Abramov, Theor. Probab. Math. Stat. 112 (2025), 1–15].
The study of recurrence or transience of Markov processes has a long history going back to the fifties of the last century. Particular Markov chains were studied by Gillis [1], Harris [2], Hodges and Rosenblatt [3], Karlin and McGregor [4], [5]. The well-known book by Chung [6] published in 1960 also discussed Markov chains with stationary transition probabilities, where the particular chain of the birth-and-death type was studied in [6].
In 1960 Lamperti [7] proved one of the most important result that establishes the conditions for recurrence or transience for the class of discrete time stochastic processes that generalizes Markov chains. His conditions of recurrence or transience required existence of the second moment of the stochastic processes, and was based on the asymptotic relationship between the first and second conditional moments of the process.
Recently Abramov [8] obtained alternative conditions for recurrence or transience of Markov chains with infinitely countable set of states. The necessary and sufficient condition obtained there was closely related to that one given for birth-and-death type Markov chains in [6]. There was no requirement on the existence of the second moments, but it was required that a Markov chain formed a connected domain. The definition of connected domain is given in [8].
The aim of this study is another look at the results of Lamperti in order to build the bridge between these two studies.
In the existing literature, we did not find alternative studies on recurrence or transience of Markov chains. For more clarity, we refer the known relatively recent books on the theory of Markov chains [9] and the stability or ergodicity of continuous time Markov processes [10].
Let \(\{X_n,\mathcal{F}_n\}\), \(\mathcal{F}_n=\sigma(X_n, X_{n-1},\ldots,X_0)\), be a nonnegative stochastic sequence satisfying the property \[\label{1} \mathsf{P}\{\limsup_{n\to\infty}X_n=\infty\}=1,\tag{1}\] (Further, the notation for stochastic sequence is provided without indication of the filtration.) The stochastic sequence \(X_n\) is said to be recurrent, if there exists \(r<\infty\) such that \(\mathsf{P}\{\liminf_{n\to\infty}X_n\leq r\}=1,\) and transient if \(\mathsf{P}\{\lim_{n\to\infty}X_n=\infty\}=1,\) e. g., [7].
Note that the definition of recurrence given here for stochastic sequences is not the same as is traditionally used for Markov chains, since it does not imply that all states are visited infinitely often. But it is Harris recurrence, where the subset \([0, r]\) is assumed to be visited infinitely often [10]. The definition of recurrence in [8] is in traditional sense. This however makes no difference, since the definition in traditional sense is implied by that given by Harris.
Assume that \[\label{2} \mathsf{E}\{|X_{n+1}-X_n|^{2+\epsilon}~|~\mathcal{F}_n\}\leq B<\infty \;\text{for some positive}\;\epsilon,\tag{2}\] and denote \[\begin{align} \overline{\mu}(x)&=&\mathrm{ess} \sup\mathsf{E}\{X_{n+1}-X_n~|~X_n=x, X_{n-1}, X_{n-2},\ldots, X_0\},\\ \underline{\mu}(x)&=&\mathrm{ess} \inf\mathsf{E}\{X_{n+1}-X_n~|~X_n=x, X_{n-1}, X_{n-2},\ldots, X_0\},\\ \overline{v}(x)&=&\mathrm{ess} \sup\mathsf{E}\{(X_{n+1}-X_n)^2~|~X_n=x, X_{n-1}, X_{n-2},\ldots, X_0\},\\ \underline{v}(x)&=&\mathrm{ess} \inf\mathsf{E}\{(X_{n+1}-X_n)^2~|~X_n=x, X_{n-1}, X_{n-2},\ldots, X_0\}. \end{align}\] (Throghout the paper, \(\epsilon\) is assumed to satisfy \(0<\epsilon\leq1\).)
The following theorem is due to Lamperti [7].
Theorem 1. (Lamperti [7].) Let the nonnegative stochastic sequence \(X_n\) satisfy 1 and 2 , and, as \(x\to\infty\), \[\overline{\mu}(x)\leq\frac{\underline{v}(x)}{2x}+O\left(\frac{1}{x^{1+\epsilon}}\right).\] Then the stochastic sequence is recurrent. If instead for some \(\theta>1\) and almost all large \(x\), \[\underline{\mu}(x)\geq\frac{\theta\overline{v}(x)}{2x},\] then the stochastic sequence is transient.
Unfortunately, this quite general Theorem 1 addresses the question about recurrence or transience for quite small class of stochastic sequences, in which \(\underline{\mu}(x)\neq\overline{\mu}(x)\) or/and \(\underline{v}(x)\neq\overline{v}(x)\) for all large \(x\). We tested the two examples considered in [8], and for both of them, despite of their clarity, Theorem 1 did not give any answer. Therefore, we will only discuss its corollary given below, the proof of which was based on Doob’s convergence theorem for semimartingales [11].
Corollary 2. (Lamperti [7].) Assume that a nonegative stochastic sequence \(X_n\) satisfies 1 , and assume that for \(x\to\infty\), we almost surely have \[\begin{align} \mu(x)&=&\mathsf{E}\{X_{n+1}-X_n~|~X_n=x, X_{n-1},\ldots,X_0\}=\frac{\xi}{2x}+O\left(\frac{1}{x^{1+\epsilon}}\right), \\ v(x)&=&\mathsf{E}\{(X_{n+1}-X_n)^2~|~X_n=x, X_{n-1},\ldots,X_0\}=r^2+O\left(\frac{1}{x^{\epsilon}}\right),\\ &&\text{where}\;\xi \;\text{and}\;r\;\text{are positive values}.\\ \end{align}\] Then the stochastic sequence is recurrent, if for almost all \(x\to\infty\), \[\mu(x)\leq \frac{v(x)}{2x}+O\left(\frac{1}{x^{1+\epsilon}}\right).\] If instead for some \(\theta>1\) and almost all large \(x\), \[\mu(x)\geq\frac{\theta v(x)}{2x},\] then the stochastic sequence is transient.
Remark 3. The specified term \(\frac{\xi}{2x}\) presented on the right-hand side of the function \(\mu(x)\) in the formulation of Corollary 2 is convenient for our purposes.
The rest of this study is organized as follows. In Section 2, we prove a new theorem that is based on Corollary 2. In Section 3, we discuss the theorem proved in Section 2 and show its connection with the main result in [8].
The conditions of recurrence and transience for Markov processes given by Corollary 2 are expressed explicitly via the limits with probability 1 of \(x\mu(x)\) and \(v(x)\) as \(x\to\infty\), and, hence, the statement of the corollary is defined for the classes of stochastic sequences where these limits exist and takes the given values.
The new theorem proved in this section suggests the representation that looks slightly more complicated than that in Corollary 2. However, it will help to understand clearer the connection with the recent result obtained in [8] as well as with the known results for birth-and-death processes expressed in the terms of intensities. This new theorem will require the existence of the limits of \(x|\mu(x)|\) and \(v(x)\) with probability \(1\), where the only limit of \(x\mu(x)\) is used explicitly, and, in addition to this, the limit with probability \(1\) of \(\nu(x) = \mathsf{E}\{|X_{n+1} -X_n|~\big|~ X_n = x\}\) is used. The advantage of this is as follows. The presence of \(\nu(x)\) has a clear physical meaning, and the representation given in the formulation of the theorem enables us to explain the connection with the result in [8] in the next section. As well, the proof provided in the paper enables us to understand a new phenomena in the behaviour of the considered stochastic sequences and Markov chains.
Before formulating and proving our result, we provide an elementary example that will help us further to understand our finding at an intuitive level.
Example 1. Let \(X_n\) be a birth-and-death chain that is defined by its transition probabilities as follows: \(P_{0,1}=1\), \(P_{i,i+1}=p_i\), \(P_{i,i-1}=q_i\), \(i\geq1\) (\(p_i+q_i=1\)). Let \(Y_n\) be the other birth-and-death type chain with the following transition probabilities: \(P_{0,2}=1\), \(P_{2i,2i+2}=p_i\), \(P_{2i,2i-2}=q_i\), \(i\geq1\). As we can see, the only difference between \(X_n\) and \(Y_n\) is in scaling parameter: the jump size of \(Y_n\) is \(2\), while the jump size of \(X_n\) is \(1\). So, considering these two processes on same probability space, one can write \(Y_n=2X_n\). Therefore, if \(X_n\) is recurrent, then so does \(Y_n\), and vice versa.
We have \[\label{3} \begin{align} \mathsf{E}[Y_{n+1}-Y_n~|~Y_n=y]&=\mathsf{E}[2X_{n+1}-2X_n~|~2X_n=y]\\ &=2\mathsf{E}\left[X_{n+1}-X_n~|~X_n=\frac{y}{2}\right]. \end{align}\tag{3}\] Now, assuming that for large \(x\) \[\mathsf{E}[X_{n+1}-X_n~|~X_n=x]= \frac{\xi}{2x}+O\left(\frac{1}{x^{1+\epsilon}}\right),\] for \(x\to\infty\), from 3 we obtain \[\mathsf{E}[Y_{n+1}-Y_n~|~Y_n=y]=2\frac{\xi}{y}+O\left(\frac{1}{y^{1+\epsilon}}\right)\] for \(y\to\infty\). Finally, substituting \(y=2x\) one can see that \[\frac{\mathsf{E}[Y_{n+1}-Y_n~|~Y_n=2x]}{\mathsf{E}[(Y_{n+1}-Y_n)^2~|~Y_n=2x]}=\frac{\mathsf{E}[X_{n+1}-X_n~|~X_n=x]}{\mathsf{E}[(X_{n+1}-X_n)^2~|~X_n=x]}+O\left(\frac{1}{x^{1+\epsilon}}\right).\]
The aforementioned example enables us to guess that the role of \(v(x)\) in the formulation of Corollary 2 is a normalization factor only. Understanding this pushes us to prove a new theorem, where the normalization factor has clearer physical interpretation.
Theorem 4. Assume that the conditions of Corollary 2 are fulfilled, and assume additionally that \(x|\mu(x)|\) with probability \(1\) converges to the finite limit as \(x\to\infty\). Then the stochastic sequence is recurrent if, as \(x\to\infty\), we almost surely have \[\frac{\mathsf{E}\{X_{n+1}-X_n~|~X_n=x, X_{n+1}>X_n, X_{n-1},\ldots, X_0\}}{\mathsf{E}\{X_{n}-X_{n+1}~|~X_n=x, X_{n+1}<X_n, X_{n-1},\ldots, X_0\}}\leq1+\frac{R}{x}+O\left(\frac{1}{x^{1+\epsilon}}\right),\] where \(R=\mathsf{E}\{|X_{n+1}-X_n|~\big|~X_{n}=x, X_{n-1},\ldots, X_0\}\). If instead \[\frac{\mathsf{E}\{X_{n+1}-X_n~|~X_n=x, X_{n+1}>X_n, X_{n-1},\ldots, X_0\}}{\mathsf{E}\{X_{n}-X_{n+1}~|~X_n=x, X_{n+1}<X_n, X_{n-1},\ldots, X_0\}}\geq1+\frac{R\theta}{x}+O\left(\frac{1}{x^{1+\epsilon}}\right),\] for some \(\theta>1\) and almost all large \(x\), then the stochastic sequence is transient.
Proof. The class of stochastic sequences \(X_n\) satisfying for almost all large \(x\) the conditions \[\begin{align} \mathsf{E}\{X_{n+1}-X_n~|~X_n=x, X_{n-1},\ldots, X_0\}&=&\frac{\xi}{2x}+O\left(\frac{1}{x^{1+\epsilon}}\right),\\ \mathsf{E}\{(X_{n+1}-X_n)^2~|~X_n=x, X_{n-1},\ldots, X_0\}&=&r^2+O\left(\frac{1}{x^{\epsilon}}\right), \end{align}\] will be denoted by \(\frak{A}_{\xi,r^2}\).
Let \(\frak{A}_{\xi,1}\) and \(\frak{A}_{r\xi,r^2}\) (with \(r\neq 1\)) be two classes of stochastic sequences. Apparently, there is one-to-one correspondence between these two classes. Indeed, if a stochastic sequence \(X_n\) belong to \(\frak{A}_{\xi,1}\), then the stochastic sequence \(Y_n=rX_n\) belongs to \(\frak{A}_{r\xi,r^2}\). Inversely, if a stochastic sequence \(Y_n\) belongs to \(\frak{A}_{\xi,r^2}\), then the stochastic sequence \(X_n=r^{-1}Y_n\) belongs to \(\frak{A}_{r^{-1}\xi,1}\). As it was noted before, the stochastic sequences \(X_n\) and \(Y_n=rX_n\) has the same classification. If one is recurrent, the same is another.
Let \(\frak{A}_{\xi,1}\) be the class of stochastic sequences with parameters \(\xi\) and \(1\), and let \(X_n\) belong to this class. Let us first find the asymptotic behavior of the ratio \[\frac{\mathsf{E}\{X_{n+1}-X_n~|~X_n=x, X_{n+1}>X_n, X_{n-1},\ldots, X_0\}}{\mathsf{E}\{X_{n}-X_{n+1}~|~X_n=x, X_{n+1}<X_n, X_{n-1},\ldots, X_0\}}\]
The assumptions that, as \(x\to\infty\), \[\label{4468} \begin{align} \mathsf{E}\{X_{n+1}-X_n~|~X_n=x, X_{n-1},\ldots, X_0\}=\frac{\xi}{2x}+O\left(\frac{1}{x^{1+\epsilon}}\right), \end{align}\tag{4}\] \[\begin{align} \mathsf{E}\{(X_{n+1}-X_n)^2~\big|~X_{n}=x, X_{n-1},\ldots, X_0\} =1+O\left(\frac{1}{x^{\epsilon}}\right), \end{align}\] generally yields \[\label{4461} \begin{align} \mathsf{P}\{X_{n+1}> X_n~|~X_n=x, X_{n-1},\ldots, X_0\}=p+\frac{pa\xi}{2x}+O\left(\frac{1}{x^{1+\epsilon}}\right), \end{align}\tag{5}\] \[\label{4462} \begin{align} \mathsf{P}\{X_{n+1}< X_n~|~X_n=x, X_{n-1},\ldots, X_0\}=q-\frac{qb\xi}{2x}+O\left(\frac{1}{x^{1+\epsilon}}\right), \end{align}\tag{6}\] where \(p+q=1\) and \(qb\geq pa+O(x^{-\epsilon})\). The relation between \(a\) and \(b\) will be specified further.
Prove 5 and 6 . Indeed, assume that given \(X_0, X_1,\ldots, X_{n-1}\) and \(X_n=x\), the difference \(X_{n+1}-X_n\) takes a negative value, zero, or positive value with some conditional probabilities \(\bar{q}_x, \bar{r}_x, \bar{p}_x\), respectively, that depend on \(x\) and the history. (To avoid unnecessary complications, we do not indicate the dependence of the history in our notation.) Since \[\mathsf{E}\{(X_{n+1}-X_n)^2~\big|~X_{n}=x, X_{n-1},\ldots, X_0\}\] is finite with probability \(1\) for all \(x\) and uniformly converges almost surely to \(1\) as \(x\to\infty\), then taking into account 4 we have the following rough estimates: \[\label{4} \begin{align} \mathsf{P}\{X_{n+1}>X_n~\big|~X_{n}=x, X_{n-1},\ldots, X_0\}=\bar{p}_x =p+O\left(\frac{1}{x^\epsilon}\right), \end{align}\tag{7}\] and \[\label{5} \begin{align} \mathsf{P}\{X_{n+1}<X_n~\big|~X_{n}=x, X_{n-1},\ldots, X_0\}=\bar{q}_x =q+O\left(\frac{1}{x^\epsilon}\right), \end{align}\tag{8}\] with almost sure uniform limits as \(x\to\infty\), where \(p\) and \(q\) are the constants, and for large \(x\), \(\bar{p}_x+\bar{q}_x=1+O\left(\frac{1}{x^\epsilon}\right)\). That is, \[\mathsf{P}\{X_{n+1}=X_n~\big|~X_{n}=x, X_{n-1},\ldots, X_0\}=\bar{r}_x=O\left(\frac{1}{x^\epsilon}\right).\]
Next, following the assumption of the existence of the limit of \(x|\mu(x)|\) with probability \(1\) as \(x\to\infty\), with the same probability we have the uniform limit of \(x\mu(x)\) as \(x\to\infty\) as well, and hence 7 and 8 can be represented in form 5 and 6 involving the additional constants \(a\) and \(b\).
Note that in the case \(qb=pa+O(x^{-\epsilon})\), for almost all large \(x\) we have \[\mathsf{P}\{X_{n+1}=X_n~|~X_n=x, X_{n-1},\ldots, X_0\}=O\left(\frac{1}{x^{1+\epsilon}}\right).\]
It follows from 5 , 6 and 4 that \[\begin{align} \mathsf{E}\{X_{n+1}-X_n~\big|~X_{n}=x, X_{n+1}>X_n, X_{n-1},\ldots, X_0\}&=&A_1+O\left(\frac{1}{x^{1+\epsilon}}\right),\\ \mathsf{E}\{X_{n+1}-X_n~\big|~X_{n}=x, X_{n+1}<X_n, X_{n-1},\ldots, X_0\}&=&-A_2+O\left(\frac{1}{x^{1+\epsilon}}\right), \end{align}\] for certain positive values \(A_1\) and \(A_2\) satisfying the property \(A_1p=A_2q+O(x^{-1-\epsilon})\). Then, \[\label{4467} \mathsf{E}\{|X_{n+1}-X_n|~\big|~X_{n}=x, X_{n-1},\ldots, X_0\}=A_1p+O\left(\frac{1}{x^{1+\epsilon}}\right).\tag{9}\]
The relation between \(a\) and \(b\) must be specified by \(A_1p(a+b)=1+O(x^{-\epsilon})\), that yields 4 by the total expectation formula. Indeed, \[\begin{align} &\mathsf{E}\{X_{n+1}-X_n~|~X_n=x, X_{n-1},\ldots, X_0\}\\ &=\mathsf{E}\{X_{n+1}-X_n~|~X_n=x, X_{n+1}>X_n, X_{n-1},\ldots, X_0\}\\ &\quad\quad\times\mathsf{P}\{X_{n+1}> X_n~|~X_n=x, X_{n-1},\ldots, X_0\}\\ &\quad+\mathsf{E}\{X_{n+1}-X_n~|~X_n=x, X_{n+1}<X_n, X_{n-1},\ldots, X_0\}\\ &\quad\quad\times\mathsf{P}\{X_{n+1} <X_n~|~X_n=x, X_{n-1},\ldots, X_0\}\\ &=A_1p\left(\frac{(a+b)\xi}{2x}\right)+O\left(\frac{1}{x^{1+\epsilon}}\right)=\frac{\xi}{2x}+O\left(\frac{1}{x^{1+\epsilon}}\right). \end{align}\]
Hence, \[\begin{align} &\frac{\mathsf{E}\{X_{n+1}-X_n~|~X_n=x, X_{n+1}> X_n, X_{n-1},\ldots, X_0\}}{\mathsf{E}\{X_{n}-X_{n+1}~|~X_n=x, X_{n+1}< X_n, X_{n-1},\ldots, X_0\}}\\ &\quad =\frac{A_1p+\frac{A_1pa\xi}{2x}+O\left(\frac{1}{x^{1+\epsilon}}\right)}{A_2q-\frac{A_2qb\xi}{2x}+O\left(\frac{1}{x^{1+\epsilon}}\right)}\\ &\quad = 1+\frac{(a+b)\xi}{2x}+O\left(\frac{1}{x^{1+\epsilon}}\right). \end{align}\]
Now, let \(s\) be such the constant that \[\mathsf{E}\{|sX_{n+1}-sX_n|~\big|~sX_{n}=x, X_{n-1},\ldots, X_0\}=1.\] In other words, the new stochastic sequence \(X_n^\prime=sX_n\) belongs to the class \(\frak{A}_{s\xi, s^2}\). Then, it is readily seen from the similar derivations provided before that \[\frac{\mathsf{E}\{X_{n+1}^\prime-X_n^\prime~|~X_n^\prime=x, X_{n+1}^\prime> X_n^\prime, X_{n-1}^\prime,\ldots, X_0^\prime\}}{\mathsf{E}\{X_{n}^\prime-X_{n+1}^\prime~|~X_n^\prime=x, X_{n+1}^\prime< X_n^\prime. X_{n-1}^\prime,\ldots, X_0^\prime\}}=1+\frac{\xi}{x}+O\left(\frac{1}{x^{1+\epsilon}}\right),\]
It follows from Corollary 2 that the stochastic sequence \(X_n^\prime\) is recurrrent, if \(\xi\leq1+O\left(\frac{1}{x^{1+\epsilon}}\right)\). If instead \(\xi\geq\theta>1\), then the stochastic sequence is transient. Furthermore, recurrence (resp. transience) of \(X_n^\prime\) implies recurrence (resp. transience) of stochastic sequence \(X_n\), and vice versa.
Indeed, for the stochastic sequence \(X_n^\prime\) we have \[\begin{align} &\mathsf{E}\{X_{n+1}^\prime-X_n^\prime~|~X_n^\prime=x, X_{n-1}^\prime,\ldots, X_0^\prime\}\\ &\quad=s\mathsf{E}\{X_{n+1}-X_n~|~X_n=s^{-1}x, X_{n-1},\ldots, X_0\}\\ &\quad=\frac{s^2\xi}{2x}+O\left(\frac{1}{x^{1+\epsilon}}\right), \end{align}\] \[\mathsf{E}\{(X_{n+1}^\prime-X_n^\prime)^2~|~X_n^\prime=x, X_{n-1}^\prime,\ldots, X_0^\prime\}=s^2+O\left(\frac{1}{x^{\epsilon}}\right),\] and the result follows.
We consider now a stochastic sequence \(Y_n=RX_{n}^\prime\) belonging to the class \(\frak{A}_{sR\xi, s^2R^2}\). Apparently, \[\begin{align} &\mathsf{E}\{|Y_{n+1}-Y_n|~\big|~Y_{n}=x, Y_{n-1},\ldots, Y_0\}\\ &\quad=\mathsf{E}\{|RX_{n+1}^\prime-RX_n^\prime|~\big|~RX_{n}^\prime=x, X_{n-1}^\prime,\ldots, X_0^\prime\}\\ &\quad=R\mathsf{E}\{|X_{n+1}^\prime-X_n^\prime|~\big|~X_{n}^\prime=R^{-1}x, X_{n-1}^\prime,\ldots, X_0^\prime\}\\ &\quad=R+O\left(\frac{1}{x^\epsilon}\right). \end{align}\]
Together with this we have: \[\begin{align} &\frac{\mathsf{E}\{Y_{n+1}-Y_n~|~Y_n=x, Y_{n+1}>Y_n, Y_{n-1},\ldots, Y_0\}}{\mathsf{E}\{Y_{n}-Y_{n+1}~|~Y_n=x, Y_{n+1}<Y_n, Y_{n-1},\ldots, Y_0\}}\\ &=\frac{\mathsf{E}\{RX_{n+1}^\prime-RX_n^\prime~|~RX_n^\prime=x, X_{n+1}^\prime>X_n^\prime, X_{n-1}^\prime,\ldots, X_0^\prime\}}{\mathsf{E}\{RX_{n}^\prime-RX_{n+1}^\prime~|~RX_n^\prime=x, X_{n+1}^\prime<X_n^\prime, X_{n-1}^\prime,\ldots, X_0^\prime\}}\\ &=\frac{\mathsf{E}\{X_{n+1}^\prime-X_n^\prime~|~X_n^\prime=R^{-1}x, X_{n+1}^\prime>X_n, X_{n-1}^\prime,\ldots, X_0^\prime\}}{\mathsf{E}\{X_{n}^\prime-X_{n+1}^\prime~|~X_n^\prime=R^{-1}x, X_{n+1}^\prime<X_n^\prime, X_{n-1}^\prime,\ldots, X_0^\prime\}}\\ &=1+\frac{R\xi}{x}+O\left(\frac{1}{x^{1+\epsilon}}\right). \end{align}\] These last derivations finish the proof. ◻
In this section, we discuss the connection of the result given by Theorem 4 and the aforementioned result in [8]. In the proof of Theorem 4, we introduced the class \(\frak{A}_{\xi, r^2}\) of processes. As it was mentioned in the proof of Theorem 4, there is one-to-one correspondence between elements of the class \(\frak{A}_{r\xi, r^2}\) and elements of the class \(\frak{A}_{\xi, 1}\). Furthermore, if a Markov chain belonging to \(\frak{A}_{r\xi, r^2}\) is recurrent (resp. transient), then the same is true for the corresponding Markov chain belonging to \(\frak{A}_{\xi, 1}\), and vice versa. Hence, one can restrict our attention by considering Markov chains belonging to the selected class \(\frak{A}_{s\xi, s^2}\) (considered in the proof of Theorem 4) that satisfies the property \[\label{d0} \mathsf{E}\{|X_{n+1}-X_n|~\big|~X_{n}=x\}=1+O\left(\frac{1}{x^\epsilon}\right).\tag{10}\]
According to Theorem 4, the class of Markov chains \(X_n\) belonging to \(\frak{A}_{s\xi,s^2}\) is recurrent, if as \(x\to\infty\), \[\label{d1} \frac{\mathsf{E}\{X_{n+1}-X_n~|~X_n=x, X_{n+1}>X_n\}}{\mathsf{E}\{X_{n}-X_{n+1}~|~X_n=x, X_{n+1}<X_n\}}\leq1+\frac{1}{x}+O\left(\frac{1}{x^{1+\epsilon}}\right).\tag{11}\] If instead \[\label{d2} \frac{\mathsf{E}\{X_{n+1}-X_n~|~X_n=x, X_{n+1}>X_n\}}{\mathsf{E}\{X_{n}-X_{n+1}~|~X_{n}=x, X_{n+1}<X_n\}}\geq1+\frac{\theta}{x}\tag{12}\] for some \(\theta>1\) and all large \(x\), then the Markov chain is transient.
Let us now consider the series \[\label{6} \sum_{i=1}^\infty\prod_{n=1}^i\frac{\mathsf{E}\{X_{n}-X_{n+1}~|~X_{n+1}<X_n\}}{\mathsf{E}\{X_{n+1}-X_{n}~|~ X_{n+1}>X_n\}}.\tag{13}\] Under a special assumption made in [8] that a Markov chain forms connected domain, it was proved that a Markov chain is recurrent, if and only if 13 diverges. Under condition 10 , the sufficient condition for divergence is 11 and for convergence is 12 . The similar more extended sufficient conditions for recurrence or transience of birth-and-death processes can be find in [12].
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