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We give Sir James Jeans's notion of 'normal state' a mathematically precisedefinition. We prove that normal cells of trajectories exist in the Hamiltonianheat-bath model of an assembly of linearly coupled oscillators that generatesthe Ornstein--Uhlenbeck process in the limit of an infinite number of degreesof freedom. This, in some special cases, verifies some far-reaching conjecturesof Khintchine on the weak ergodicity of a dynamical system with a large numberof degrees of freedom. In order to estimate the theoretical auto-correlationfunction of a time series from the sample auto-correlation function of one ofits realisations, it is usually assumed without justification that the timeseries is ergodic. Khintchine's conjectures about dynamical systems with largenumbers of degrees of freedom justifies, even in the absence of ergodicity,approximately the same conclusions.  Para emplear el correlograma de los valores muestrales de un procesoestoc\'astico para estimar su funci\'on te\'orica de autocorrelaci\'on, porregla general se asume, sin justificaci\'on, que el proceso es erg\'odico. Peroen 1943, Khintchine conjetur\'o proposiciones de gran importancia en esteasunto, que justificar\'i an una aproximaci\'on a las mismas estimaciones a\'unsin la ergodicidad del sistema. Mostraremos casos particulares de lasconjeturas de Khintchine para asambleas de osciladores lineales.

Some Special Cases of Khintchine's Conjectures in Statistical Mechanics:  Approximate Ergodicity of the Auto-Correlation Functions of an Assembly of  Linearly Coupled Oscillators