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We study the power density spectrum (PDS) of light curves of the observedgamma-ray bursts (GRBs) to search for a direct signature for cosmological timedilation in the PDS statistics with the GRBs whose redshifts $z$'s are known.The anticorrelation of a timescale measure and a brightness measure is indirectevidence of its effect. On the other hand, we directly demonstrate that a timedilation effect can be seen in GRB light curves. We find that timescales tendto be shorter in bursts with small redshift, as expected from cosmologicaltime-dilation effects, and we also find that there may be non-cosmologicaleffects constituting to this correlation. We discuss its implication oninterpretations of the PDS analysis results. We put forward another caution tothis kind of analysis when we statistically exercise with GRBs whose $z$ isunknown.Comment: 12 pages, 2 figures, 2 tables, accepted to ApJ

Fourier Analysis of Gamma-Ray Burst Light Curves: Searching for a Direct Signature of Cosmological Time Dilation