Analyses of mean daily discharges and annual peak discharges for streams in 14 of the United States and spanning a wide range of climates show that frequency of occurrence relationships for the large-discharge tails of both follow power laws. The number N(Q) of days on which the discharge exceeds Q, or the number of years in which the peak discharge exceeds Q, is related to Q by N(Q) $propto$ Q-α. Values of the exponent α (1 < α < 6) decrease in magnitude with increasing aridity so that the ratio of the frequency of occurrence of very large discharges to that of smaller discharges is higher in arid than in humid environments of the United States. To examine the effect of climate change on bed load transport and river incision, we obtain a curve-fit relating mean annual discharge per unit area of drainage basin (an effective precipitation rate $overline{P}$) to α: α - 1 $propto$ $overline{P}$n, where n ≈ 1.6. Using this relationship, we confirm that rivers in arid regions should incise less rapidly as climate becomes yet more arid. (If no water flows, the "river" transports no sediment.) Whether aridification of an initially humid environment leads to increased or decreased incision rates, however, depends on the minimum (threshold) discharge capable not only of transporting bed load but also sufficient to scour alluvium from the riverbed and then erode the bedrock. The curve fit relating $overline{P}$ to α implies that for aridification to accelerate incision, floods that recur only once or twice per millennium (or less frequently) must carry out most of the incision. An overestimate of n could permit smaller, more frequent floods to incise, but it appears that in only special circumstances will aridification accelerate steam incision.