Central Terms of Markov Walks

Abstract

A $\{0, 1\}$-valued discrete time stochastic process $\beta = \{\beta_n\}^\infty_{n=1}$ will be referred to simply as a walk. The notion of central (modal) term of a binomial distribution is generalized to the conditional-on-the-past distributions of $N$th partial sums of walks. The emphasis here is placed on the smallest possible central term $V_A(N)$ within a given class $A$ of walks. If $A$ consists of (i) all walks, (ii) all stationary independent walks, (iii) all stationary Markov walks which are invariant under interchange of 0 and 1, then, respectively, (i) $\{N \cdot V_A(N)\}^\infty_{N=1}$, (ii) $\{N^{\frac{1}{2}} \cdot V_A(N)\}^\infty_{N=1}$, (iii) $\{N \cdot V_A(N)/(\log N)^{\frac{1}{2}}\}^\infty_{N=2}$ are bounded sequences which are bounded away from zero.