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Suppose that in a ballot candidate $A$ scores $a$ votes and candidate $B$ scores $b$ votes and that all possible $\left(a+ba\right)$ voting sequences are equally probable. Denote by ${\alpha }_r$ and by ${\beta }_r$ the number of votes registered for $A$ and for $B$, respectively, among the first $r$ votes recorded, $r=1,\cdots ,a+b$. The purpose of this paper is to derive, for $a\ge b-c$, the probability distributions of the random variables defined as the number of subscripts $r=1,\cdots ,a+b$ for which (i) ${\alpha }_r={\beta }_r-c$, (ii) ${\alpha }_r={\beta }_r-c$ but ${\alpha }_{r-1}={\beta }_{r-1}-c\pm 1$, (iii) ${\alpha }_r={\beta }_r-c$ but ${\alpha }_{r-1}={\beta }_{r-1}-c\pm 1$ and ${\alpha }_{r+1}={\beta }_{r+1}-c\pm 1$, where $c=0,\pm 1,\pm 2,\cdots$.

The contents of the paper is concerned with the two-sample problem where $F_m\left(x\right)$ and $G_n\left(x\right)$ are two empirical distribution functions. The difference $F_m\left(x\right)-G_n\left(x\right)$ changes only at an $x_i,i=1,2,...,m+n$, corresponding to one of the observations. Let $R_{mn}^{+}\left(j\right)$ denote the subscript $i$ for which $F_m\left(x_i\right)-G_n\left(x_i\right)$ achieves its maximum value $D_{mn}^{+}$ for the $j$th time $(j=1,2,...)$. The paper deals with the probabilities for $R_{mn}^{+}\left(j\right)$ and for the vector $(D_{mn}^{+},R_{mn}^{+}\left(j\right))$ under $H_0:F=G$, thus generalizing the results of Steck-Simmons (1973). These results have been derived by applying the random walk model.