The Annals of Mathematical Statistics

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

J. H. B. Kemperman

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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.

Article information

Source
Ann. Math. Statist., Volume 30, Number 2 (1959), 448-462.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177706262

Digital Object Identifier
doi:10.1214/aoms/1177706262

Mathematical Reviews number (MathSciNet)
MR102883

Zentralblatt MATH identifier
0092.36402

JSTOR
links.jstor.org

Citation

Kemperman, J. H. B. Asymptotic Expansions for the Smirnov Test and for the Range of Cumulative Sums. Ann. Math. Statist. 30 (1959), no. 2, 448--462. doi:10.1214/aoms/1177706262. https://projecteuclid.org/euclid.aoms/1177706262


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