  We study a plug in least squares estimator for the change point parameter
where change is in the mean of a high dimensional random vector under
subgaussian or subexponential distributions. We obtain sufficient conditions
under which this estimator possesses sufficient adaptivity against plug in
estimates of mean parameters in order to yield an optimal rate of convergence
$O_p(\xi^{-2})$ in the integer scale. This rate is preserved while allowing
high dimensionality as well as a potentially diminishing jump size $\xi,$
provided $s\log (p\vee T)=o(\surd(Tl_T))$ or $s\log^{3/2}(p\vee
T)=o(\surd(Tl_T))$ in the subgaussian and subexponential cases, respectively.
Here $s,p,T$ and $l_T$ represent a sparsity parameter, model dimension,
sampling period and the separation of the change point from its parametric
boundary. Moreover, since the rate of convergence is free of $s,p$ and
logarithmic terms of $T,$ it allows the existence of limiting distributions.
These distributions are then derived as the {\it argmax} of a two sided
negative drift Brownian motion or a two sided negative drift random walk under
vanishing and non-vanishing jump size regimes, respectively. Thereby allowing
inference of the change point parameter in the high dimensional setting.
Feasible algorithms for implementation of the proposed methodology are
provided. Theoretical results are supported with monte-carlo simulations.
