The Limit of a Ratio of Convolutions

Abstract

Let $\{X_i\}, i = 1, 2, \cdots$ be a sequence of nonnegative, independent, identically distributed random variables, and let $F_n(x) = P\{X_1 + \cdots + X_n < x\}$. The purpose of this note is to investigate $\lim_{n\rightarrow\infty} \lbrack F_n(a_1)/F_n(a_2)\rbrack,$ where $0 < a_1 < a_2 < \infty$. In order to do so, estimates are required of the extreme lower tail of the distribution $F_n(x)$, for large $n$. There are a number of known results on the probability in the tail of a convolution (see [1], [2], [3], [4]), but none are appropriate for the present problem. Let $X$ denote a typical $X_i$. In the case when $P\{0 < X < x_0\} = 0$ for some $x_0 > 0$, the required limit is trivial to evaluate. This result is stated in Theorem 1. In the contrary case it is shown in Theorem 3 that a sufficient condition for $\lbrack F_n(a_1)/F_n(a_2)\rbrack \rightarrow 0$ is that there exist real $\gamma \geqq 0$ and $k > 0$ such that \begin{equation*}\tag{1}0 < \lim_{x\rightarrow0+}x^{-\gamma} P\{0 < X < x\} = k < \infty.\end{equation*} This result is achieved by means of an estimate of $F_n(x)$ which is given in Theorem 2. The Condition (1) is satisfied by a wide class of random variables including those with density functions $f(x)$ such that for some $\alpha \geqq 0, 0 < \lim_{x\rightarrow 0+} x^{-\alpha}f(x) = k < \infty.$ It is easy to verify that most "textbook" densities satisfy this condition, as do all densities with a positive right-continuous $k$th derivative $(0 \leqq k < \infty)$ at zero. Although (1) is not necessary for $\lbrack F_n(a_1)/F_n(a_2)\rbrack \rightarrow 0$ (a class of examples will be given later to illustrate this point), one would expect that some regularity condition near the origin would be required. In attempting to weaken the sufficient Condition (1), one might conjecture that conditions such as the existence of a bounded density $f(x)$, such that $0 < f(x)$ on an open interval $(0, \epsilon), 0 < \epsilon$, is sufficient for $\lbrack F_n(a_1)/F_n(a_2)\rbrack \rightarrow 0$. This conjecture is at the present neither proved, nor disproved by a counterexample.