Conditions for the weak convergence of distributions of separable statistics

Divisible statistics

“…We note that in the case of symmetric DS one can take as the number 7 any number from the interval [0, I]; in [86][87][88][89][90][91][92][93][94]112] ?----|. Applying the theorem of [34], we get the following assertion.…”

“…qn=(q(n I) ..... ,it2, ,..+,> W+,n~) We note that by virtue of the continuity of the distribution density defined by (1.3), the set V n is open. Continuous versions of the regular conditional distribution were used in Holst [86][87][88][89]93], Holst and Rao [96,97], and in [30,31,[34][35][36][37].…”

“…~ we get a representation (1.15). Instead of Condition 1.8' we require that the following three restrictions hold [34]. We note that in the case of symmetric DS one can take as the number 7 any number from the interval [0, I]; in [86][87][88][89][90][91][92][93][94]112] ?----|.…”

“…~ is absolutely integrable with respect to the argument (s, t) on the set Rl~ XAm (cf. ( 9) of [34]) so that for each value n j>nx the distribution density of the r.v. Nn with respect to the product of s Lebesgue measures and the corresponding counting measure exists:…”