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Limit theorems for conditional distributions. (English. Russian original) Zbl 0821.60038
Discrete Math. Appl. 4, No. 6, 519-542 (1994); translation from Diskretn. Mat. 6, No. 4, 107-132 (1994).
Let $$\{(X_{mN_ n}, Y_{mN_ n})\}$$, $$m = 1, \dots, N_ n$$, $$n = 1,2, \dots,$$ be a sequence of independent in each $$n$$th series random vector from $$R^ 2$$, $$X = \sum^{N_ n}_{m = 1} X_{mN_ n}$$, $$Y = \sum^{N_ n}_{m = 1} Y_{mN_ n}$$, and let $$Y$$ be an integer- valued random variable. The author obtains the remainder term in the central limit theorems and large deviations estimates for the conditional distribution $P \bigl( X < x \sqrt{ \text{var} (x)} + E(x) \mid Y = y \bigr) = P_{N_ n} (x/y)$ as $$n$$ and $$N$$ tend to infinity.

##### MSC:
 60F05 Central limit and other weak theorems 60F10 Large deviations
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