Return magnitudes can be dependent across time even when signed returns are uncorrelated, so absence of linear return correlation does not establish independence.

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A 10% fall yesterday may well increase the odds of another 10% move today, but provide no advanced way of telling whether it will be up, or down. If so, the correlation vanishes, in spite of the strong dependence. Large price changes, tend to be followed by more large changes, positive or negative. Small changes, tend to be followed by more small changes. Volatility clusters. What use is that? Plenty, if you are in the business of managing, avoiding, or profiting from risk. A bank is required by its regulators to estimate the value of its market assets daily and set aside a certain amount of capital as a cushion against loss. A better, more accurate way of estimating those potential losses would save the bank money and the financial system grief. A fund manager or investor who cannot tolerate the risk of a large loss might, when the financial storm signs are up, simply trim his sales and avoid old bets. And options traders strive to profit from risk.

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Third is a statement of normality that all the price changes taken together, from small to large, vary in accordance with the mild bell curve distribution. In my view, that is two claims too many. The first, though not proven by data, is at least not much contradicted by it, And it certainly helps, in an intuitive way, to explain why we so often guess the market wrong. But the others are simply false. The data overwhelmingly show that the magnitude of price changes depends on those of the past, and that the bell curve is a nonsense. Speaking mathematically, markets can exhibit dependence without correlation. The key to this paradox lies in the distinction between the size and the direction of price changes. Suppose that the direction is uncorrelated with the past. The fact that prices fell yesterday does not make them more likely to fall today. It remains possible for the absolute changes to be dependent.