Measuring bias in cyclic random walks (Q387426)

Let \(X\) be a random variable taking two values with probabilities \(p\) and \(1-p\). Then the bias \(B(X)\) of \(X\) is defined by \(B(X)=|p-q|\). The authors explore the notion of the bias for investigation of convergence of sums modulo 2. Namely, they obtain that if \(X_1, X_2, \dots \) is a sequence of independent coin-flips, then NEWLINE\[NEWLINE X_1+ \dots + X_n \mod 2 NEWLINE\]NEWLINE converges in measue to \(\mu\) (\(\mu\) is the probability distribution of a fair coin) as \(n\to\infty\) if and only if NEWLINE\[NEWLINE \lim\limits_{n\to\infty}\prod_{i=1}^nB(X_i)=0. NEWLINE\]NEWLINE This is generalized for sums modulo \(m\), \(m\geq 2\). Let \(\mathbb{Z}_m=\{0, 1, \dots , m-1\}\), and let a random variable \(X\) take values in \(\mathbb{Z}_m\) with distribution \({\mathbf p}=(p(0), p(1), \dots , p(m-1))\). The authors define five types of biases for the distribution of \(X\) and study their properties. One of those biases is defined by NEWLINE\[NEWLINE B_4({\mathbf p})={1\over 2}\max\limits_{r\in \mathbb{Z}_m}\sum_{l\in \mathbb{Z}_m}\left|p(l)-p(l-r)\right|, NEWLINE\]NEWLINE and is applied for the study of the asymptotic behaviour of sums modulo \(m\). Let \(X_1, X_2, \dots\) be a sequence of independent random variables on \(\mathbb{Z}_m\) with distributions \({\mathbf p}_1, {\mathbf p}_2, \dots\), respectively, and \(\mu\) be the uniform distribution on \(\mathbb{Z}_m\). Then the authors obtain that NEWLINE\[NEWLINE X_1+ \dots + X_n \mod m NEWLINE\]NEWLINE converges in measure to \(\mu\) as \(n\to\infty\) if and only if NEWLINE\[NEWLINE \lim\limits_{n\to\infty}\prod_{i=1}^nB_4({\mathbf p}_i)=0. NEWLINE\]