Suppose the return vectors R1,R2,... are independent and identically distributed. Let w be a positive constant. Assume maxπ E[log(π

Rt)] > −∞ and let π∗ be a solution to maxπ E[log(π

Rt)].

Let W∗ be the wealth process defined by the intertemporal budget constraint

(8.1) with πt = π∗ and Yt = Ct = 0 for each t and W∗

0 = w. Consider any other portfolio π for which E[log(π

Rt))] < maxπ E[log(π

Rt)].

Let W bethe wealth process defined bythe intertemporal budget constraint (8.1)

withπt = π and Yt = Ct = 0 for each t andW0 = w. Showthat, with probability 1, there exists T (depending on the state of the world) such that W∗

t > Wt for all t ≥ T. Hint: Apply the strong law of large numbers to (1/T)logW∗

T and to

(1/T)logWT.