 Solution

Consider an occurrence counted by $G_m$. Before the next counted occurrence, the exploration must first take a downward step of the simple symmetric random walk, which has probability $1/2$, and then choose the decrement $-1$ among the three equally likely values of $\xi$, which has probability $1/3$. Thus it terminates the visits to the current record value with probabilityThe Strong Markov property at successive counted occurrences makes these trials independent. Therefore $G_m$ has the geometric distribution on $\{1,2,\dots \}$ with parameter $1/6$, andfor every $j\ge 1$.