  This paper considers a sequence of discrete-time random walk markets with a
safe and a single risky investment opportunity, and gives conditions for the
existence of arbitrages or free lunches with vanishing risk, of the form of
waiting to buy and selling the next period, with no shorting, and furthermore
for weak convergence of the random walk to a Gaussian continuous-time
stochastic process. The conditions are given in terms of the kernel
representation with respect to ordinary Brownian motion and the discretisation
chosen. Arbitrage and free lunch with vanishing risk examples are established
where the continuous-time analogue is arbitrage-free under small transaction
costs - including for the semimartingale modifications of fractional Brownian
motion suggested in the seminal Rogers (1997) article proving arbitrage in fBm
models.
