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Probabilities of medium deviations in limit theorems for conditional distributions of sums of independent random variables. (Russian) Zbl 0758.60025
Let $$\{(X_{m,N_ n},Y_{m,N_ n})$$, $$1\leq m\leq N_ n\}$$, $$n\geq 1$$, be a triangular array of $$R^ 2$$-valued random vectors independent within each row, such that $$Y_{N_ n}=\sum_{m=1}^{N_ n} Y_{m,N_ n}$$ are integer valued. The asymptotic behaviour of probabilities $P_{N_ n}(x\mid y)=P[X_{N_ n}<x\sqrt{DX_{N_ n}}+E X_{N_ n}\mid Y_{N_ n}=y],$ is investigated, where $$X_{N_ n}=\sum_{m=1}^{N_ n}X_{m,N_ n}$$. Let $\rho_{N_ n}=\text{corr}(X_{N_ n},Y_{N_ n}), \qquad U(N_ n,y)=(y-E Y_{N_ n})/\sqrt{DY_{N_ n}},$ $x_ \mp=x/\sqrt{1-\rho_{N_ n}^ 2} \mp U(N_ n,y)\cdot\rho_{N_ n}/\sqrt{1-\rho_{N_ n}^ 2},$ and let $$\Phi_{N_ n}(x\mid y)$$ denote the normal distribution function with mean $$U(N_ n,y)\cdot\rho_{N_ n}$$ and variance $$1- \rho_{N_ n}^ 2$$. The main theorem states that if $$x=x(n)\geq 0$$ and $$y=y(n)$$ are such that $$x_ -^ 2+U^ 2(N_ n,y)\leq\delta\ln N_ n$$ and $$U^ 2(N_ n,y)\leq\delta_ 0\ln N_ n$$ with $$0\leq\delta_ 0<\min\{1,\delta\}$$ for some $$\delta>0$$, then under suitable additional assumptions concerning variances and moments of order $$2+\delta$$ for $$X_{N_ n}$$, $$Y_{N_ n}$$, and characteristic functions of $$Y_{m,N_ n}$$, we have $1-P{N_ n}(x\mid y)=(1-\Phi_{N_ n}(x\mid y))(1+o(1)),$ and if $$x_ -$$ is replaced by $$x_ +$$, then $P_{N_ n}(-x\mid y)=\Phi_{N_ n}(-x\mid y)(1+o(1))\text{ as }n\to\infty.$

##### MSC:
 60F10 Large deviations