Asymptotic Expansions for the Smirnov Test and for the Range of Cumulative Sums

Abstract

Let $z_n$ denote the position at time $n$ of a particle describing a one-dimensional random walk, such that the increments $\zeta_n = z_n - z_{n-1} (n = 1, 2, \cdots)$ are independent random variables, assuming only the values +1 and -1, each with probability $\frac{1}{2}$. Of considerable importance in many applications is the conditional probability $$p_n(i, j, c) = P(z_n = j, 0 < z_m < c, m = 1, \cdots, n \mid z_0 = i);$$ here, $i, j, c, n$ denote positive integers. In section 1, an asymptotic development for $p_n(i, j, c)$ is given; for each positive integer $m$, it yields an approximation to $p_n(i, j, c)$ with error smaller than $Cn^{-m}$ where $C$ is independent of $i, j, c$ and $n$. As a simple application, an asymptotic development for the binomial coefficient $\binom{n}{s}$ is derived by letting $i, j, c$ tend to infinity in such a manner that $j - i = 2s - n$. As a second application, an asymptotic expansion is derived for the joint distribution of the extrema of the difference between the empirical distributions of two samples of size $n$. The above asymptotic development for $p_n(i, j, c)$ is obtained by applying the central Lemma 4 to an exact formula for $p_n(i, j, c)$. In Section 5, using this formula, an exact formula is obtained for the distribution of the range $R_n$ of the $n + 1$ numbers $z_0, \cdots, z_n$. Applying Lemma 4 to it, a complete asymptotic expansion for the distribution of $R_n$ is derived.