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Consider a branching process $\{Z_n\}_{n\ge 0}$ with immigration in varying environment. For $a\in\{0,1,2,...\},$ let $C=\{n\ge0:Z_n=a\}$ be the collection of times at which the population size of the process attains level $a.$ We give a criterion to determine whether the set $C$ is finite or not. For critical Galton-Watson process, we show that $|C\cap [1,n]|/\log n\rightarrow S$ in distribution, where $S$ is an exponentially distributed random variable with $P(S>t)=e^{-t},\ t>0.$, Comment: 20 pages

In this paper, we study a transient spatially inhomogeneous random walk with asymptotically zero drifts on the lattice of the positive half line. We give criteria for the finiteness of the number of points having exactly the same local time and/or upcrossing time and weak cutpoints (a point $x$ is called a weak cutpoint if the walk never returns to $x-1$ after its first upcrossing from $x$ to $x+1$). In addition, for the walk with some special local drifts, we also give the order of the expected number of these points in $[1,n].$ Finally, we show that, when properly scaled, the number of these points in $[1,n]$ converges in distribution to a random variable with the standard exponential distribution. Our results answer three conjectures related to the local time, the upcrossing time, and the weak cutpoints proposed by E. Cs\'aki, A. F\"oldes, P. R\'ev\'esz [J. Theoret. Probab. 23 (2) (2010) 624-638]., Comment: In this version, we answer two more problems proposed by E. Cs\'aki, A. F\"oldes, P. R\'ev\'esz [J. Theoret. Probab. 23 (2) (2010) 624-638]. The first version considered only the number of points with exactly the same local time

In this paper, we study (1,2) and (2,1) random walks in varying environments on the lattice of positive half line. We assume that the transition probabilities at site $n$ are asymptotically constants as $n\rightarrow\infty.$ For (1,2) random walk, we get some elaborate asymptotic behaviours of various escape probabilities and hitting probabilities of the walk. Such observations and some delicate analysis of continued fractions and the product of nonnegative matrices enable us to give criteria for finiteness of the number of cutpoints of both (1,2) and (2,1) random walks, which generalize E. Cs\'aki, A. F\"oldes and P. R\'ev\'esz [J. Theor. Probab. 23: 624-638 (2010)] and H.-M. Wang [Markov Processes Relat. Fields 25: 125-148 (2019)]. For near-recurrent random walks, whenever there are infinitely many cutpoints, we also study the asymptotics of the number of cutpoints in $[0,n].$, Comment: 31 pages

In this paper, we consider certain linear-fractional branching processes with immigration in varying environments. For $n\ge0,$ let $Z_n$ counts the number of individuals of the $n$-th generation, which excludes the immigrant which enters into the system at time $n.$ We call $n$ a regeneration time if $Z_n=0.$ We give first a criterion for the finiteness or infiniteness of the number of regeneration times. Then, we construct some concrete examples to exhibit the strange phenomena caused by the so-called varying environments. It may happen that the process is extinct but there are only finitely many regeneration times. Also, when there are infinitely many regeneration times, we show that for each $\varepsilon>0,$ the number of regeneration times in $[0,n]$ is no more than $(\log n)^{1+\varepsilon}$ as $n\rightarrow\infty.$, Comment: 12 pages

In this paper we study a 2-type linear-fractional branching process in varying environment with asymptotically constant mean matrices. Let $\nu$ be the extinction time and for $k\ge1$ let $M_k$ be the mean matrix of offspring distribution of individuals of the $(k-1)$-th generation. Under certain conditions, we show that $P(\nu=n)$ and $P(\nu>n)$ are asymptotically equivalent to some functions of products of spectral radii of the mean matrices. This paper complements a former result [arXiv: 2007.07840] which requires in addition a condition $\forall k\ge1,\rm{det}(M_k)<-\varepsilon$ for some $\varepsilon>0.$ Such a condition excludes a large class of mean matrices. As byproducts, we also get some results on asymptotics of products of nonhomogeneous matrices which have their own interests., Comment: 22 pages

Consider two-type linear-fractional branching processes in varying environments with asymptotically constant mean matrices. Let $\nu$ be the extinction time. Under certain conditions, we show that both $P(\nu=n)$ and $P(\nu>n)$ are asymptotically the same as some functions of the products of spectral radii of the mean matrices. We also give an example for which $P(\nu=n)$ decays with various speeds such as $\frac{c}{n(\log n)^2},$ $\frac{c}{n^\beta},\beta >1$ et al. which are very different from the ones of homogeneous multitype Galton-Watson processes., Comment: 29 pages; In this version, we prove the results for general mean matrices and some technical conditions are removed

Let $A_kA_{k-1}\cdots A_1$ be product of some nonnegative 2-by-2 matrices. In general, its elements are hard to evaluate. Under some conditions, we show that $\forall i,j\in\{1,2\},$ $(A_kA_{k-1}\cdots A_1)_{i,j}\sim c\varrho(A_k)\varrho(A_{k-1})\cdots \varrho(A_1)$ as $k\rightarrow\infty,$ where $\varrho(A_n)$ is the spectral radius of the matrix $A_n$ and $c\in(0,\infty)$ is some constant, so that the elements of $A_kA_{k-1}\cdots A_1$ can be estimated. As applications, consider the maxima of certain excursions of (2,1) and (1,2) random walks with asymptotically zero drifts. We get some delicate limit theories which are quite different from the ones of simple random walks. Limit theories of both the tail and critical tail sequences of continued fractions play important roles in our studies., Comment: 30 pages;In the second version, we add the result of (1,2) random walk

Consider a nearest-neighbor random walk with certain asymptotically zero drift on the positive half line. Let $M$ be the maximum of an excursion starting from $1$ and ending at $0.$ We study the distribution of $M$ and characterize its asymptotics, which is quite different from those of simple random walks., Comment: 10 pages

In this paper, we consider Galton-Watson processes with immigration. Pick $i(\ge2)$ individuals randomly without replacement from the $n$-th generation and trace their lines of descent back in time till they coalesce into $1$ individual in a certain generation, which we denote by $X_{i,1}^n$ and is called the coalescence time. Firstly, we give the probability distribution of $X_{i,1}^n$ in terms of the probability generating functions of both the offspring distribution and the immigration law. Then by studying the limit behaviors of various functionals of the Galton-Watson process with immigration, we find the limit distribution of $X_{2,1}^n$ as $n\rightarrow\infty.$, Comment: 12 pages; We add some new result in this version, which studies the limit distribution of the coalescence time, see Theorem 2 of the paper

Consider a transient near-critical (1,2) random walk on the positive half line. We give a criteria for the finiteness of the number of the skipped points (the points never visited) by the random walk. This result generalizes (partially) the criteria for the finiteness of the number of cutpoingts of the nearest neighbor random walk on the line by Cs\'aki, F\"olders, R\'ev\'esz [J Theor Probab (2010) 23: 624-638].

Consider $(1,2)$ random walk in random environment $\{X_n\}_{n\ge0}.$ In each step, the walk jumps at most a distance $2$ to the right or a distance $1$ to the left. For the walk transient to the right, it is proved that almost surely $\lim_{x\rightarrow\infty}\frac{\#\{X_n:\ 0\le X_n\le x,\ n\ge0\}}{x}=\theta$ for some $0<\theta<1.$ The result shows that the range of the walk covers only a linear proportion of the lattice of the positive half line. For the nearest neighbor random walk in random or non-random environment, this phenomenon could not appear in any circumstance., Comment: 14 pages

In this paper, we study a birth and death process $\{N_t\}_{t\ge0}$ on positive half lattice, which at each discontinuity jumps at most a distance $R\ge 1$ to the right or exactly a distance $1$ to the left. The transitional probabilities at each site are nonhomogeneous. Firstly, sufficient conditions for the recurrence and positive recurrence are presented. Then by the branching structure within random walk with one-side bounded jumps set up in Hong and Wang (2013), the explicit form of the stationary distribution of the process $\{N_t\}_{t\ge0}$ is formulated., Comment: 6 pages

Let $\omega=(\omega_i)_{i\in\mathbb Z}=(\mu^{L}_i,...,\mu^{1}_i,\lambda_i)_{i\in \mathbb Z}$, which serves as the environment, be a sequence of i.i.d. random nonnegative vectors, with $L\ge1$ a positive integer. We study birth and death process $N_t$ which, given the environment $\omega,$ waits at a state $n$ an exponentially distributed time with parameter $\lambda_n+\sum_{l=1}^L\mu^{l}_n$ and then jumps to $n-i$ with probability ${\mu^i_n}/(\lambda_n+\sum_{l=1}^L\mu^{l}_n),$ $i=1,...,L$ or to $n+1$ with probability ${\lambda_n}/(\lambda_n+\sum_{l=1}^L\mu^{l}_n).$ A sufficient condition for the existence, a criterion for recurrence, and a law of large numbers of the process $N_t$ are presented. We show that the first passage time $T_1\overset{\mathscr D}{=}\xi_{0,1}+\sum_{i\le -1}\sum_{k=1}^{U_{i,1}}\xi_{i,k}+\sum_{i\le -1}\sum_{k= 1}^{U_{i,1}+...+U_{i,L}}\tilde{\xi}_{i+1,k},$ where $(U_{i,1},...,U_{i,L})_{i\le0}$ is an $L$-type branching process in random environment and, given $\omega,$ $\xi_{i,k},\ \tilde\xi_{i,k},\ i\le 0,\ k\ge 1$ are mutually independent random variables such that $P_\omega(\xi_{i,k}\ge t)=e^{-(\lambda_i+\sum_{l=1}^L\mu^{l}_i)t},\ t\ge 0.$ This fact enables us to give an explicit velocity of the law of large numbers., Comment: 9 pages

We study random walk with unbounded jumps in random environment. The environment is stationary and ergodic, uniformly elliptic and decays polynomially with speed $Dj^{-(3+\varepsilon_0)}$ for some small $\varepsilon_0>0$ and proper $D>0.$ We prove a law of large number with positive velocity under the condition that the annealed mean of the hitting time of the positive half lattice is finite. Secondly, we consider birth and death process with bounded jumps in stationary and ergodic environment. Under the uniformly elliptic condition, we prove a law of large number and give the explicit formula of its velocity., Comment: 25 pages