Consider the Dickey–Fuller statistic

. It is well known (see, e.g., Phillips, 1987) that if the data-generating process (d.g.p.) is the random walk X_t = X_t−1 + ε_t, with initial condition X₀ = 0 and with {ε_t} i.i.d. N(0,σ²), then

where B is a standard Brownian motion and

denotes weak convergence with respect to the uniform metric.

Show that the result (1) holds even if the random walk {X_t} is constrained to be nonnegative according to the following d.g.p. (the so-called random walk with one reflecting barrier; see Cox and Miller, 1965)

with initial condition X₀ = 0.