Two sided bounds of rate convergence in limit theorem for minimum of random vectors

In this article upper and low bounds of a rate convergence for minimums of independent and identically distributed random (i.i.d.r.) vectors are constructed. These bounds have common power and different logarithmical multiplyers. An interest to this problem is called by following causes. At first I. Siganov obtained upper bounds for minimums of i.i.d.r. variables, which may be considered as a foundation for two sided bounds. At second last years P. Rocchi constructed new models of a life-time for biological objects, which are based on stochastic entropy methods and give distributions analogous to considered ones. At third in mathematical statistics and reliability theory there are so called Marshall-Olkin distributions, which may be interpreted as limit distributions for minimums of i.i.d.r. vectors. This interpretation widens a class of Marshall-Olkin distributions.