  We investigate the statistics of the gap, G_n, between the two rightmost
positions of a Markovian one-dimensional random walker (RW) after n time steps
and of the duration, L_n, which separates the occurrence of these two extremal
positions. The distribution of the jumps \eta_i's of the RW, f(\eta), is
symmetric and its Fourier transform has the small k behavior 1-\hat{f}(k)\sim|
k|^\mu with 0 < \mu \leq 2. We compute the joint probability density function
(pdf) P_n(g,l) of G_n and L_n and show that, when n \to \infty, it approaches a
limiting pdf p(g,l). The corresponding marginal pdf of the gap, p_{\rm gap}(g),
is found to behave like p_{\rm gap}(g) \sim g^{-1 - \mu} for g \gg 1 and 0<\mu
< 2. We show that the limiting marginal distribution of L_n, p_{\rm time}(l),
has an algebraic tail p_{\rm time}(l) \sim l^{-\gamma(\mu)} for l \gg 1 with
\gamma(1<\mu \leq 2) = 1 + 1/\mu, and \gamma(0<\mu<1) = 2. For l, g \gg 1 with
fixed l g^{-\mu}, p(g,l) takes the scaling form p(g,l) \sim g^{-1-2\mu} \tilde
p_\mu(l g^{-\mu}) where \tilde p_\mu(y) is a (\mu-dependent) scaling function.
We also present numerical simulations which verify our analytic results.
