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Recent standardization of portions of the millimeter-wave (mm-wave) band for fifth-generation (5G) operation has called for further research on how short-term fading behaves in that unexplored part of the spectrum. With such a target, this paper reports on a thorough measurement campaign conducted in an indoor environment characterized by rich-multipath scattering, a part of a modern building, with floor and ceiling constructed of reinforced concrete over steel plates with wood and plasterboard-paneled walls. Particularly, measurements have been performed in a variety of scenarios, under line-of-sight (LoS) and non-line-of-sight (nLoS) conditions, for a wide range of frequencies, namely from 25 to 40 GHz- a span of 15 GHz- therefore, including 26, 28 and 39 GHz. First and second order statistics of representative fading models, namely Rayleigh, Rice, Nakagami, folded normal, <inline-formula> <tex-math notation="LaTeX">${\alpha } $ </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">${\mu } $ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">${\eta } $ </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">${\mu }$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">${\kappa } $ </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">${\mu } $ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">${\alpha } $ </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">${\eta } $ </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">${\kappa } $ </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">${\mu } $ </tex-math></inline-formula> have been investigated. The metrics used in the analysis were the normalized mean square error (NMSE), the Kolmogorov-Smirnov (KS), and the Akaike information criterion (AIC). Additionally, the study of the <inline-formula> <tex-math notation="LaTeX">${\kappa } $ </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">${\mu } $ </tex-math></inline-formula> model is advanced, in which new, exact, simple closed-form expressions for probability density function, cumulative distribution function, and level crossing rate are derived for some particular cases, namely for <inline-formula> <tex-math notation="LaTeX">${\mu = n+ 1/2}$ </tex-math></inline-formula> in which <inline-formula> <tex-math notation="LaTeX">${n\in {{\mathbb {N}}}}$ </tex-math></inline-formula>.