Open Access
June, 1959 Asymptotic Expansions for the Smirnov Test and for the Range of Cumulative Sums
J. H. B. Kemperman
Ann. Math. Statist. 30(2): 448-462 (June, 1959). DOI: 10.1214/aoms/1177706262


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.


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J. H. B. Kemperman. "Asymptotic Expansions for the Smirnov Test and for the Range of Cumulative Sums." Ann. Math. Statist. 30 (2) 448 - 462, June, 1959.


Published: June, 1959
First available in Project Euclid: 27 April 2007

zbMATH: 0092.36402
MathSciNet: MR102883
Digital Object Identifier: 10.1214/aoms/1177706262

Rights: Copyright © 1959 Institute of Mathematical Statistics

Vol.30 • No. 2 • June, 1959
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