Fractional derivatives and integrals are convolutions with a power law. Multiplying by an exponential factor leads to tempered fractional derivatives and integrals. Tempered fractional diffusion equations, where the usual second derivative in space is replaced by a tempered fractional derivative, govern the limits of random walk models with an exponentially tempered power law jump distribution. The limiting tempered stable probability densities exhibit semi-heavy tails, which are commonly observed in finance. Tempered power law waiting times lead to tempered fractional time derivatives, which have proven useful in geophysics. The tempered fractional derivative or integral of a Brownian motion, called a tempered fractional Brownian motion, can exhibit semi-long range dependence. The increments of this process, called tempered fractional Gaussian noise, provide a useful new stochastic model for wind speed data. A tempered difference forms the basis for numerical methods to solve tempered fractional diffusion equations, and it also provides a useful new correlation model in time series.

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We propose a new time series model for velocity data in turbulent flows. The new model employs tempered fractional calculus to extend the classical 5/3 spectral model of Kolmogorov. Application to wind speed and water velocity in a large lake are presented, to demonstrate the practical utility of the model.

Tempered fractional Brownian motion is obtained when the power law kernel in the moving average representation of a fractional Brownian motion is multiplied by an exponential tempering factor. This paper develops the theory of stochastic integrals for tempered fractional Brownian motion. Along the way, we develop some basic results on tempered fractional calculus.

We discuss invariance principles for autoregressive tempered fractionally integrated moving averages in α-stable (1 < α ≤ 2) i.i.d. innovations and related tempered linear processes with vanishing tempering parameter λ ∼ λ * /N . We show that the limit of the partial sums process takes a different form in the weakly tempered (λ * = 0), strongly tempered (λ * = ∞), and moderately tempered (0 < λ * < ∞) cases. These results are used to derive the limit distribution of the OLS estimate of AR(1) unit root with weakly, strongly, and moderately tempered moving average errors.Keywords: invariance principle; tempered linear process; autoregressive fractionally integrated moving average; tempered fractional stable/Brownian motion; tempered fractional unit root distribution;An important example of such processes is the two-parametric class ARTFIMA(0, d, λ, 0) of tempered fractionally integrated processes, generalizing the well-known ARFIMA(0, d, 0) class, written as 1) is the backward shift. Due to the presence of the exponential tempering factor e −λk the series in (1.1) and (1.5) absolutely converges a.s. under general assumptions on the innovations, and defines a strictly stationary process. On the other hand, for λ = 0 the corresponding stationary processes in (1.1) and (1.5) exist under additional conditions on the parameter d. See Granger and Joyeux [12], Hosking [13], Brockwell and Davis [5], Kokoszka and Taqqu [15]. We also note (see e.g. [10], Ch. 3.2) that the (untempered) linear process X d,0 of (1.1) with coefficients satisfying (1.2) for 0 < d < 1/2 is said long memory, while (1.2) and (1.3) for −1/2 < d < 0 is termed negative memory and (1.4) short memory, respectively, parameter d usually referred to as memory parameter.