will always periodogram produce correct frequencies? let say we have  some data,some signal written in this form

$x(t)=s(t)+\epsilon(t)$

where  $s(t)$ is pure signal and  $\epsilon(t)$ is  white noise  with mean zero and some variance  ,question is that if we have enough sample data,should  periodogram always  estimate frequencies  correctly?let say we have following sinusoidal data

$y[t]=A_1(sin(\omega_1*t+\phi_1)+A_2*sin(\omega_2*t+\phi_2)+....+A_p*sin(\omega_p*t+\phi_p)$+$z(t)$

i know that there is problem of variance increasing related to periodogram,but will it affect on frequency estimation?also i know that  resolution of periodogram is $1/N$,but let us suppose that  two frequency are not so close to each other,then will periodogram produce correct frequency estimation?variance is not important for me know
A periodogram created from a finite number of samples using an DFT is not really a "correct" frequency estimator, as much as it is a (leaky) frequency energy binning sieve, where the width of each bin is set by the DFT length and the window function applied.  

The width and spacing of each bin may or may not be narrow enough for you to consider an error of up to half that size to still result in a useful frequency estimate.  Various interpolation methods that post-process DFT results may allow more precise estimates of spectral peaks than the bin spacing.