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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 \[ P_{N_ n}(-x\mid y)=\Phi_{N_ n}(-x\mid y)(1+o(1))\text{ as }n\to\infty. \]

60F10 Large deviations