arXiv subjects
Publications and source records attributed to Vladimir Rotar.
We consider a limit theorem for a triangular array of point processes generated by non-identically distributed random variables, and apply the result for the analysis of the limiting behavior of the Argmaximum of independent random variables, as well as for some step processes.
We consider a limit theorem for the distribution of a r.v. $Y_n:=argmax {\{X_i, i= 1,..., n\}},$ where $X_i'$s are independent continuous non-negative random variables. The r.v.'s $\{X_i, i=1,..., n\}$, may be interpreted as the gains of $n$ players in a game, and the r.v. $Y_n$ itself as the number of a ``winner". In the case of i.i.d.r.v.'s, the distribution of $Y_n$ is, clearly, uniform on $\{1,..., n\},$ while when the $X'$s are non-identically distributed, the problem requires some calculations.
We consider the invariance principle without the classical condition of asymptotic negligibility of individual terms. More precisely, we explore the difference of the following two distributions in the space C (of continuous functions on [0,1]). The first is the distribution of the continuous piecewise linear partial-sum process generated by a sequence of independent random variables, and the second is the distribution of the similar process generated by the sequence of normal r.v.'s with the same first two moments. The novelty is that the condition of negligibility of the r.v.'s is not imposed. We establish a necessary and sufficient condition of the weak convergence of the difference mentioned to zero measure in C.