Some Properties of a Trinomial Random Walk Conditioned on End Points

Annual Meeting in Mathematics 2023

Abstract

Given a sequence of trinomial random variables $\displaystyle\{ X_i \}^\infty_{i=1}$ and define $S_n =\sum_{i=1}^n X_i$ and $S_0 = 0$, we study some properties of $X_i $ conditioned on $S_n = 0.$ The mathematical expressions of expectation, variance and covariance were investigated. We found that the a finite sequence $(X_1, X_2, \ldots, X_n)$ conditioned on $S_n = 0$ is exchangeable. Moreover, the expectation of $X_i$ is zero and the covariance of $X_i$ and $X_j$ where $i \neq j$ is nonpositive. Furthermore, we extend the previous setting to a rescaled trinomial random walk. Some properties on the extension were derived.