random walks with stochastically bounded increments

YOKOHAMA MATHEMATICAL JOURNAL VOL. 42, 1994

RANDOM WALKS WITH STOCHASTICALLY BOUNDED INCREMENTS: RENEWAL THEORY VIA FOURIER ANALYSIS By

GEROLD ALSMEYER (Received

Summary.

March 23, 1993)

Random walks with stochastically bounded increments have been introduced in [2], [3] as natural generalizations of those with .i.d. increments. In this article we present Blackwell-type renewal theorems proved by means of Fourier analysis. In the special case of in, dependent these results lead to generalizations of earlier ones in the literature, notably in [3] where proofs were based on coupling technique which is a purely probabilistic device. As a further aPplication we prove Blackwell’s renewal theorem for certain random walks with stationary 1dependent increments that appear in Markov renewal theory as subsequences of Markov random walks. $X_{0},$

$X_{1},$

$S_{N}=(S_{n})_{n\geqq 0}$

$\cdots$

$i$

$X_{0},$

$X_{1}$

$\cdot$

1. Introduction Random walks with stochastically lower $and/or$ upper bounded increments, see Definition 1.1 below, are a natural generalization of those with i.i. . increments and have been introduced in [2], [3]. Certain drift bounds describing the mean growth of these random walks over finite remote time intervals as well as related characterization results are given in [2], whereas [3] is devoted to the proof of Blackwell-type renewal theorems under appropriate additional assumptions. Of principal importance there is the use of the coupling method, a probabilistic device which has regained great importance since the seventies. this article we will derive Blackwell-type renewal theorems via the more classical approach based upon Fourier analysis. We keep the basic notation of [2] and [3] which is briefly summarized below. Let be a sequence of real-valued, integrable random variables on a probability space with associated random walk , defined through $S_{n}=X_{0}+\cdots+X_{n}$ for all $n\in N$ Let be an arbitrary filtration to $d$

$\ln$

$X_{N}=(X_{n})_{n\geqq 0}$

$(\Omega, \mathcal{F}, P)$

$S_{N}$

$\mathcal{F}_{N}$

1991 Mathematics Subject Classification: Primary $60G50,60J15$

.

$60K05$

; Secondary $60G40,60G42$ ,

Key words and phrases: Random walk, stochastic boundedness, stochastic stability, maximal minorant, minimal majorant, Blackwell’s renewal theorem, sequences of type AC and CC, Markov renewal theory, Fourier analysis.

2

G. ALSMEYER

which , i.e. the canonical filtration of is adapted and . $F$ For each measure , we use the same letter for its “distribution function” and thus write $F(t)$ for $F((-\infty, t$ ]). If $F$ is a probability measure, let $\overline{F}(t)=1-F(t)$ $k\in N$ and , we and $\mu(F)$ its mean value provided it exists. For further define $X_{N}$

$\mathcal{G}_{n}=\sigma(X_{0}, X_{n})$

$X_{N}$

$\mathcal{G}_{N}$

$0\leqq j\leqq n$

$n,$

$S_{n.k}=S_{n+k}-S_{n}$

$L_{0}=X_{0}$

,

,

$m_{n+1}=E(X_{n+1}|\mathcal{F}_{n})$

$L_{n+1}=m_{1}+\cdots+m_{n+1}$ ,

$L_{n.k}^{f}=E(S_{n.l}|\mathcal{F}_{j})=E(L_{n.i}|\mathcal{F}_{j})$

$Q_{n}(dx)=P(S_{n}\in dx)$ ,

,

$L_{n,k}=L_{n+k}-L_{n}$

,

,

$Q_{n.k}(\omega, dx)=P(S_{n.k}\in dx|\mathcal{F}_{n})(\omega)$

,

is chosen to be a regular conditional distribution. Finally, $B$ always where the supremum norm on the vector denotes the $Borel-\sigma- field$ over $R$ and $Q_{n.k}$

$\Vert\cdot\Vert_{\infty}^{\backslash }$

space

$L_{\infty}(\Omega, \mathcal{F}, P)$

.

Definition 1.1. A sequence -stochastically bounded $(s.b.)w$ .

$X_{N}$

$r$

. . $t$

, adapted to a filtration , if for distributions

$\mathcal{F}_{n}$

$\mathcal{F}_{N}$

, is called $G$

$F,$

with finite

means (A.1)

$G(t)\leqq Q_{n.1}(\cdot, t)\leqq F(t)$

-stochastically stable w.r.t. (A.2)

$\mathcal{F}_{N}$

a.s. for all

, if it is s.b.

and

$t\in R$

w.r. . $t$

$\mathcal{F}_{N}$

$n\in N$

.

and if additionally

$\lim_{k\rightarrow\infty}snp||k^{-1}L_{n.k}^{n}-\theta\Vert_{\infty}=0n\geqq 0$

which is then called the mean of . $w.r.t$ , . if -ultimately stochastically bounded for is s.b. $w$ .r.t. and . is then called an entrance -time , such that some . time of , if it is ultimately s.b. with a sequence –ultimately stochastically stable $w.r.t$ . of entrance times such that

holds for some

$\theta\in R$

$X_{N}$

$X_{\tau+N}$

$\mathcal{F}_{N}$

$\mathcal{F}_{N}$

$ E\taux)dx$

.

The remaining calculations are then done analogously, first for bounded $X$, and an obvious

by splitting up the range of integration of the last integral above in manner. We do not supply the details again.

Lemma 5.4. If

$Y_{N+1}$

is non-reducible and tight, then there are

$a,$

$T>0$

such

that (5.9)

$1-\varphi_{\max}(t)\geqq at^{2}$

for all

Proof. Let us consider the sequence by . As already mentioned, whence $|\varphi_{n}|^{2},$

$n\geqq 1$

$t\in(-T, T)$

and

whose associated F.t. are given is also non-reducible and tight

$Y_{N+1}^{l}$

$Y_{N+1}^{l}$

$F(t)=\sup_{n\geqq\iota}P(Y_{n}^{l}\leqq t)def$

.

$G(t)=\inf_{n\geqq 1}P(Y_{n}^{\epsilon}\leqq t)def$

G. ALSMEYER

20

are proper distribution functions reducibility and symmetry).

Now

(by tightness) and satisfy $F(O)