 Solution

Financial data show approximate, scale-dependent regularities rather than an exact universal scaling law. Heavy return tails can resemble a power law over an intermediate range, but the fitted exponent, cutoff and range depend on instrument, period and sampling. A straight segment on a log-log plot alone is not enough to establish a power law.

For independent finite-variance increments, variance grows linearly with horizon and typical fluctuations scale as $\Delta t^{1/2}$. Central-limit aggregation can make longer-horizon returns more Gaussian even when short-horizon returns are heavy-tailed. Real data can depart from a single self-similar distribution because of volatility clustering, changing regimes, asymmetric responses, intraday seasonality and market microstructure. Apparent different scaling of higher moments may reflect intermittent volatility, but moments estimated from rare tails can also be unstable or may not exist in an idealized model.

Useful tests compare rescaled distributions, structure functions and volatility correlations across several scales, with uncertainty estimates and checks for nonstationarity. Scaling is a valuable approximate modelling guide over a documented range, not proof that one exponent describes every asset and horizon or that historical tail behaviour fixes future extreme losses.