Limit Theorems for Stopped Random Walks

Abstract

In this paper we prove several asymptotic results about the expected length of time until first passage of a vector valued random walk. We suppose $X_i = (X^1_i, \cdots, X^k_i), i \geqq 1$, is a sequence of independently and identically distributed random vectors. We assume throughout that \begin{equation*}\tag{1}\max_{1 \leqq j \leqq k}E|X^j_1| < \infty; \mu_j = EX^j_1, 1 \leqq j \leqq k; \end{equation*} \\ \begin{equation*} \text{and that} 0 < \min_{1 \leqq j \leqq k} \mu_j.\end{equation*} We consider the following kind of first passage problem. Let $p(\cdot, \cdots, \cdot)$ be a real valued function of $k$ variables. Given $c > 0$ define a stopping variable $N(c) \geqq 1$ to be the least integer $n$ such that \begin{equation*}\tag{(2)}p \big (\sum^n_{i=1} X^1_i, \cdots, \sum^n_{i=1} X^k_i\big) > c,\end{equation*} with $N(c) = \infty$ if the inequality (2) fails for all $n \geqq 1$. The main result of this paper is: THEOREM 1. Suppose the function $p(\cdot, \cdots, \cdot)$ is a homogeneous function of degree one. Given the above assumptions, with probability one $\lim_{c\rightarrow\infty} c^{-1}N(c) = 1/p(\mu_1, \cdots, \mu_k)$. Suppose there is a real number $\alpha > 0$ such that \begin{equation*}\tag{3}\text{if} \min_{1 \leqq j \leqq k} a_j \geqq 0 \text{then} \alpha p(a_1, \cdots, a_k) \geqq \min_{1\leqq j \leqq k} a_j.\end{equation*} Then \begin{equation*}\tag{4} \lim_{c\rightarrow\infty}E|c^{-1} N(c) - 1/p(\mu_1, \cdots, \mu_k)| = 0.\end{equation*} If one views Theorem 1 in the light of Doob [2] Theorem 1 becomes very plausible. It is natural to define a continuous parameter process \begin{equation*}\tag{5}X(c) = c - p\big(\sum^{N(c) -1}_{i=1} X^1_i, \cdots, \sum^{N(c)-1}_{i=1} X^k_i\big).\end{equation*} In case the random variables $X^1_1, \cdots, X^k_1$ are nonnegative, using additional assumptions about $p(\cdot, \cdots, \cdot)$ it can be shown that asymptotically as $c \rightarrow \infty$ the process $X(\cdot)$ becomes a stationary Markov process. We plan to examine this question further in another place. The language of this paper is that of random variables and probability. We use freely the idea of stopped random walks. This makes it necessary to define, in the sequel, many different stopping variables. To simplify notation we use the same notation for different stopping variables and leave it to the context to distinguish the various random variables. Theorem 1 is a corollary of Theorem 3, stated below. The sequence of the argument is to prove a one-dimensional result, Theorem 2, then to prove by mathematical induction a $k$-dimensional result, Theorem 3, for the choice $p(x_1, \cdots, x_k) = \min_{1 \leqq j \leqq k} x_j$. The statements of Theorem 2 and Theorem 3 follow. THEOREM 2. Suppose $\{X_i, i \geqq 1\}$ is a sequence of independently and identically distributed random variables such that $E|X_1| < \infty \text{and} EX_1 = \mu > 0.$ If $c > 0$ is a real number define an integer valued random variable $N(c)$ to be the least integer $n \geqq 1$ such that $X_1 + \cdot + X_n > c$, with $N(c) = \infty$ if for all $n \geqq 1, X_1 + \cdot + X_n \leqq c$. Let $h(\cdot)$ be a real valued continuous strictly increasing convex function defined on $\lbrack 0, \infty)$ such that $h(0) = 0$ and $\lim_{x\rightarrow\infty} h(x)/x = \infty$. Assume that $Eh(|X_1|) \leqq \Delta < \infty$. Then $\mathrm{(6)}P(N(c) < \infty) = 1; EN(c) < \infty;$ $c \leqq \mu EN(c) \leqq c + h^{-1}(\Delta EN(c)).$ Convex functions of the type mentioned in the statement of Theorem 2 always exist. We discuss this question briefly at the beginning of Section 2. THEOREM 3. Suppose $\{X_i, i \geqq 1\}$ is a sequence of independently and identically distributed $k$-dimensional random vectors satisfying (1). Let $p(x_1, \cdots, x_k) = \min_{1 \leqq j \leqq k} x_j$ and let $N(c)$ be defined as in (2). Then $\mathrm{(7)}P(N(c) < \infty) = 1; EN(c) < \infty;$ $\lim_{c\rightarrow\infty} c^{-1}EN(c) = 1/(\min_{1 \leqq j \leqq k}\mu_j).$` We prove Theorem 2 in Section 2 for the sake of completeness and in order to have a result including a uniformity condition. Theorem 3 does not have any uniformity condition. Theorem 4 is a partial generalization of Theorem 2 in regard to a uniformity condition. THEOREM 4. Assume the hypotheses and definitions of Theorm 3. In addition assume that if $2 \leqq j \leqq k, i \geqq 1$, then $X^j_i \geqq 0$, and that $0 < \mu_1 \leqq \mu_2 \leqq \cdot \leqq \mu_k$. Let $h(\cdot)$ be a real valued nonnegative continuous function defined on $\lbrack 0, \infty)$ which satisfies \begin{equation*}\tag{(8)}h(0) = 0; \lim_{x\rightarrow\infty}h(x) /x = \infty.\end{equation*} There exists a function $a_k(\cdot, \cdot, \cdot)$ of three variables (and also depending on $h(\cdot)$ such that if $\mu > 0, \Delta > 0$ then $\lim_{c\rightarrow\infty} a_k(c, \mu, \Delta) = 0$ and if \begin{equation*}\tag{(9)}\max_{1 \leqq j \leqq k} Eh(|x^j_1|) \leqq \Delta, \min_{1 \leqq j \leqq k}EX^j_1 \geqq \mu_1,\end{equation*} then \begin{equation*}\tag{(10)}1/\mu_1 \leqq c^{-1}EN(c) \leqq 1/\mu_1 + a_k(c, \mu_1, \Delta).\end{equation*} The statement of Theorem 4 is asymmetric in that we require $X^2_1, \cdots, X^k_1$ to be nonnegative random variables. We believe that this restriction can be removed. But we have not been able to do so using the methods of this paper. The possibility of proving Theorems 2 and 4 (thereby obtaining results with uniformity conditions) was suggested to the author by arguments in a paper by Kiefer and Sacks [4]. Both Theorems have application to the study of the asymptotic behavior of the expected sample size in sequential tests. In regard to Theorem 2, since $\lim_{x\rightarrow\infty} h(x)/x = \infty$, if $d > 0$ then $\lim_{x\rightarrow\infty} h^{-1} (dx)/x = d \lim_{y\rightarrow\infty} h^{-1}(y)/y = d \lim_{z\rightarrow\infty} z/h(z) = 0$. Therefore from (6) it follows that $\lim_{c\rightarrow\infty} c^{-1} EN(c) = \mu^{-1}$. This part of Theorem 2 is known. The results of this paper on the length of time to first passage are related to the problems of renewal theory. We refer the reader to Smith [6] for a summary of results. The only multidimensional result, along the lines of this paper, in print, seems to be that of Chung [1]. In he proofs we use freely the idea of a stopped random walk and in connection with this an identity due to Wald [7]. Because this identity is used repeatedly we state it here. Suppose $\{Z_n, n \geqq 1\}$ is a sequence of independently and identically distributed random variables and $E|Z_1| < \infty$. Suppose $N$ is a random variable taking only positive integer values and $EN < \infty$. Suppose for each integer $n \geqq 1$ the event $\{N \geqq n\}$ is independent of the random variables $\{Z_m, m \geqq n\}$. Then $E\|Z_1 + \cdots + Z_N\| < \infty \text{and} E(Z_1 + \cdots + Z_N) = (EN)(EZ_1)$. Throughout the remainder of this paper we refer to this as Wald's identity and omit further mention of the reference. In the proofs given below many of the arguments require the definition of new stopping variables. For the sake of brevity the definitions of these stopping variables omit the statement that the value is $\infty$ if the stated condition fails to hold for any positive integer $n$. In this context the meaning of statements like "$P(N < \infty) = 1$" is that the exceptional set in question has measure zero. The proof of Theorem 2 is given in Section 2. The proofs of Theorem 3 and Theorem 4 require several lemmas proven in Section 3. The proof of Theorem 3 is given in Section 4, of Theorem 4 in Section 5, and of Theorem 1 in Section 6. The author is indebted to W. L. Smith for the present form of the statement of Lemma 3 and also for a very much shorter proof.