Number of appearances of events in random sequences: a new generating function approach to Type II and Type III runs

Abstract

Distributions of runs of length at least k (Type II runs) and overlapping runs of length k (Type III runs) are derived in a unified way using a new generating function approach. A new and more compact formula is obtained for the probability mass function of the Type III runs.

Appendix: Proof of Lemma 1

In this Appendix, we give the proof of Lemma 1.

Proof

Using the Wilf–Zeilberger method (Petkovsěk et al.1996), we can obtain the following linear recurrence equation for $S_m$:

$$\begin{aligned}&(m-v+3) (m-u+2) (uv -u + 2m+2) S_{m+1} \\&\quad = (m+1 )(uv - u + 2n + 4) (m-u-v) S_{m} . \end{aligned}$$

To get explicit form of $S_m$, first assume that $v-2 \ge u-1$. From the recurrence we have

$$\begin{aligned} S_{m+1} \!=\! \frac{(-1)^{m-v+3} \left[ (m+1) \cdots \right] \left[ (u+2) \cdots (u+v-m)\right] ( uv -u \!+\! 2n + 4 )}{\left[ (m-u+2) \cdots \right] \left[ (m-v+3) \cdots 1 \right] (uv + 2v - u -2)} S_{v-2} . \end{aligned}$$

When $m=v-2$, there is only one term in the sum, which leads to

$$\begin{aligned} S_{v-2} = \left( {\begin{array}{c}v-1\\ u\end{array}}\right) (-1)^{v-2} . \end{aligned}$$

After substitution of $S_{v-2}$ and rearrangements, the identity in Eq. (8) of Lemma 1 is obtained for the case when $v-2 \ge u-1$ and $m \ge 0$. If we assume $v-2 < u-1$, the same result is obtained for $m \ge 0$. In this case the recurrence ends with $m=u-1$, and we use the identity

$$\begin{aligned} S_{u-1} = \left( {\begin{array}{c}u+1\\ v\end{array}}\right) (-1)^{u-1} \end{aligned}$$

to get the explicit form of $S_m$.

The case of $m=-1$ is trivial since in this case the only value u can take is $u=0$; hence, the sum involves only one term when $r=-1$. This further restricts the values of v, which can only take $v=0$ or $v=1$ for the sum to take nonvanishing value. $\square$