  We propose a new class of mappings, called Dynamic Limit Growth Indices, that
are designed to measure the long-run performance of a financial portfolio in
discrete time setup. We study various important properties for this new class
of measures, and in particular, we provide necessary and sufficient condition
for a Dynamic Limit Growth Index to be a dynamic assessment index. We also
establish their connection with classical dynamic acceptability indices, and we
show how to construct examples of Dynamic Limit Growth Indices using dynamic
risk measures and dynamic certainty equivalents. Finally, we propose a new
definition of time consistency, suitable for these indices, and we study time
consistency for the most notable representative of this class -- the dynamic
analog of risk sensitive criterion.
