Related papers: A Proof of Basic Limit Theorem of Renewal Theory

We prove several forms of renewal theorem tailored to renewal processes with marks and clusters. In particular, for an i.i.d. sequence $(\xi_i,X_i)_{i \geq 0}$, where $\xi_0$ denotes a finite point process on $\mathbb{R}$ and $X_0$ denotes…

This paper presents a new proof of the renewal theorem by bijecting a general point process to a deterministic one (where the time between events is always fixed). It also provides insight into the workings of the renewal theorem.

In this paper we prove the following renewal-type limit theorem. Given an irrational $\alpha$ in (0,1) and R>0, let $q_{n_R}$ be the first denominator of the convergents of $\alpha$ which exceeds R. The main result in the paper is that the…

We study i.i.d. sums $\tau_k$ of nonnegative variables with index $0$: this means $\mathbf{P}(\tau_1=n) = \varphi(n) n^{-1}$, with $\varphi(\cdot)$ slowly varying, so that $\mathbf{E}(\tau_1^\varepsilon)=\infty$ for all $\varepsilon>0$. We…

Using partition generating function techniques, we prove $q$-series analogues of a formula of Frobenius generalizing Abel's convergence theorem for complex power series. Frobenius' result states that for $|q|<1$, $\lim_{q\to…

We investigate a connection between generalized Fibonacci numbers and renewal theory for stochastic processes. Using Blackwell's renewal theorem we find an approximation to the generalized Fibonacci numbers. With the help of error estimates…

For a numerical sequence ${a_n}$ satisfying broad assumptions on its "behaviour on average" and a random walk $S_n=\xi_1 +...+\xi_n$ with i.i.d. jumps $\xi_j$ with positive mean $\mu$, we establish the asymptotic behaviour of the sums…

The classical Poisson theorem says that if $\xi_1,\xi_2,...$ are i.i.d. 0--1 Bernoulli random variables taking on 1 with probability $p_n\equiv \la/n$ then the sum $S_n=\sum_{i=1}^n\xi_i$ is asymptotically in $n$ Poisson distributed with…

Let $(\xi_k,\eta_k)_{k\in\mathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $T:=(T_k)_{k\in\mathbb{N}}$ defined by…

Let $\rho$ be a borelian probability measure on $\mathrm{SL}_d(\mathbb{R})$. Consider the random walk $(X_n)$ on $\mathbb{R}^d\setminus\{0\}$ defined by $\rho$ : for any $x\in \mathbb{R}^d\setminus\{0\}$, we set $X_0 =x$ and $X_{n+1} =…

Let $a_0=b_0=0$ and $0<a_1\leq b_1<a_2\leq b_2<\ldots\leq b_{n}$ be integers. Let $Q\left(x;\bigcup_{j=1}^{n}[a_j,b_j]\right)$ be the number of integers between $1$ and $x$ such that all exponents in their prime factorization are in…

We consider a renewal process which models a cumulative shock model that fails when the accumulation of shocks up-crosses a certain threshold. The ratio limit properties of the probabilities of non-failure after n cumulative shocks are…

The Index Conjecture in zero-sum theory states that when $n$ is coprime to $6$ and $k$ equals $4$, every minimal zero-sum sequence of length $k$ modulo $n$ has index $1$. While other values of $(k,n)$ have been studied thoroughly in the…

We extend Goldie's (1991) Implicit Renewal Theorem to enable the analysis of recursions on weighted branching trees. We illustrate the developed method by deriving the power tail asymptotics of the distributions of the solutions R to: R =_D…

Renewal theorems are developed for point processes with interarrival times $W_n=\xi(X_{n+1}X_n\cdots)$, where $(X_n)_{n\in\mathbb Z}$ is a stochastic process with finite state space $\Sigma$ and $\xi\colon\Sigma_A\to\mathbb R$ is a H\"older…

Non-linear renewal theory is extended to include random walks perturbed by both a slowly changing sequence and a stationary one. Main results include a version of the Key Renewal Theorem, a derivation of the limiting distribution of the…

We consider the branching process in random environment $\{Z_n\}_{n\geq 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We focus…

The purpose of this note is to prove the celebrated Discrete Renewal Theorem in a common special case. We use only very elementary methods from real analysis, rather than markov chain theory, complex analysis, or generating functions.…

Given a zero-free region and an averaged zero-density estimate over all Dirichlet $L$-functions modulo $q\in\mathbb{N}$, we refine the error terms of the prime number theorem in all and almost all short arithmetic progressions. For example,…

It is shown that the first $n$ prime numbers $p_1,...,p_n$ determine the next one by the recursion equation $$ p_{n+1} =\lim\limits_{s\to +\infty} [\prod\limits^n_{k=1} (1-\frac{1}{p^s_k}) \sum\limits^\infty_{j=1} \frac{1}{j^s} -1]^{-1/s}.…