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This paper considers a sequence of discrete-time random walk markets with asafe and a single risky investment opportunity, and gives conditions for theexistence of arbitrages or free lunches with vanishing risk, of the form ofwaiting to buy and selling the next period, with no shorting, and furthermorefor weak convergence of the random walk to a Gaussian continuous-timestochastic process. The conditions are given in terms of the kernelrepresentation with respect to ordinary Brownian motion and the discretisationchosen. Arbitrage and free lunch with vanishing risk examples are establishedwhere the continuous-time analogue is arbitrage-free under small transactioncosts - including for the semimartingale modifications of fractional Brownianmotion suggested in the seminal Rogers (1997) article proving arbitrage in fBmmodels.

On free lunches in random walk markets with short-sale constraints and  small transaction costs, and weak convergence to Gaussian continuous-time  processes