Limit theorems for conditional distributions. (English. Russian original) Zbl 0821.60038

Mirakhmedov, Sh. A.

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.

Reviewer: A.K.Basu (Calcutta)

MSC:

60F05	Central limit and other weak theorems
60F10	Large deviations

Keywords:

rate of convergence; remainder term in the central limit theorems; large deviations estimates; conditional distribution