the Skorokhod topology on D[0,1]
/ skuh-ruh-KHOD /
Limit theorems for processes with jumps, partial sums of heavy-tailed variables converging to a stable Levy process, queueing and renewal processes, live on the space D[0,1] of cadlag functions (right-continuous with left limits). The natural metric here cannot be uniform distance: a path that jumps at time t and an approximation that jumps at t + epsilon are uniformly far apart no matter how small epsilon is, even though they are 'morally' close. The Skorokhod topology fixes this by allowing a small wiggle in time as well as in space.
D[0,1] is the set of cadlag functions x: [0,1] -> R. The Skorokhod J1 topology declares x_n -> x if there exist continuous strictly increasing time changes lambda_n: [0,1] -> [0,1] with lambda_n(0) = 0, lambda_n(1) = 1, such that BOTH sup_t |lambda_n(t) - t| -> 0 (the time distortion vanishes) AND sup_t |x_n(lambda_n(t)) - x(t)| -> 0 (after the time change, the paths are uniformly close). Intuitively, you are allowed to slightly speed up or slow down the clock to line up the jumps before comparing heights. With a cleverly chosen complete metric (Billingsley's d^0, which also controls the time-changes), D[0,1] becomes a Polish space, so all the machinery, Prokhorov, Skorokhod representation, the continuous mapping theorem, applies.
The key point is that on the subspace C[0,1] of continuous paths the Skorokhod topology coincides with the uniform topology, so nothing is lost for continuous limits, but D[0,1] additionally accommodates jump limits. The honest subtleties: addition is NOT continuous on D[0,1] (two sequences each converging in J1 can have non-converging sums if their jump times approach the same instant from different sides), so you cannot freely add cadlag limits; and J1 is only one of several Skorokhod topologies (J1, J2, M1, M2), with J1 the standard one but M1 sometimes needed when many small jumps coalesce into one big jump in the limit.
Let x_n = 1_{[1/2 + 1/n, 1]} and x = 1_{[1/2, 1]}, two cadlag step functions jumping near t = 1/2. In uniform distance ||x_n - x||_infinity = 1 for all n (they never agree near the jump). But under J1, take lambda_n mapping 1/2 to 1/2 + 1/n: after this tiny time change the jumps coincide and the paths match exactly, so x_n -> x in the Skorokhod topology.
A shifting jump: uniformly far but Skorokhod-close, because a small time reparametrization aligns the jumps.
Addition is NOT continuous on D[0,1] under J1, so you cannot routinely add Skorokhod-convergent sequences. And the limit of cadlag functions whose jumps all converge to one point may have a single jump (use M1, not J1, there).