  In this note, we introduce the notion of $\alpha$-IDT processes which is
obtained from a slight and fundamental modification of the IDT property.
Several examples of $\alpha$-IDT processes are given and Gaussian processes
which are $\alpha$-IDT are characterized. A kind example of this Gaussian
$\alpha$-IDT is the standard fractional Brownian motion. Also, we invest some
links between the $\alpha$-IDT property, with selfdecomposability, temporal
selfdecomposability, stability and self similarity.
