## An inequality for the Lévy distance between two distribution functions and its applications.(Ukrainian, English)Zbl 1224.60170

Teor. Jmovirn. Mat. Stat. 81, 53-64 (2009); translation in Theory Probab. Math. Stat. 81, 59-70 (2010).
The authors deal with the Lévy distance $${\mathcal L}(F,G)$$ between two distribution functions $$F$$ and $$G$$ defined as $${\mathcal L}(F,G)=\inf\{h: G(x-h)-h\leq F(x)\leq G(x+h)+h\text{ holds for all }x\}$$. The Lévy distance is much less popular in probability theory than the uniform distance $$\Delta(F,G)=\sup_{x\in\mathbb R}| F(x)-G(x)|$$. Note that always $${\mathcal L}(F,G)\leq\Delta(F,G)$$. The advantage of the Lévy distance becomes evident if one considers the weak convergence $$F_n\overset {w}{} G$$ as $$n\to\infty$$, which is equivalent to $${\mathcal L}(F_n,G)\to0$$ as $$n\to\infty$$ (see, for example, [B. V. Gnedenko and A. N. Kolmogorov, Limit distributions for sums of independent random variables. Cambridge: Addison-Wesley Publishing Company (1954; Zbl 0056.36001)]). If $$G$$ is continuous, then the weak convergence $$F_n\overset {w}{} G$$ as $$n\to\infty$$ is equivalent to $$\Delta(F_n,G)\to0$$ as $$n\to\infty$$; however, this property may fail if $$G$$ has discontinuities. The authors first prove the following theorem which is a generalization of the corresponding result published by K.-H. Indlekofer and O. I. Klesov [Int. J. Pure Appl. Math. 47, No. 2, 235–241 (2008; Zbl 1165.60013)].
Theorem 1. Let $$F$$ be an arbitrary distribution function and let $$\Phi$$ be the distribution function of the standard $$N(0, 1)$$ Gaussian law. Denote by $$L={\mathcal L}(F,\Phi)$$ the Lévy distance between $$F$$ and $$\Phi$$. Suppose the moment of order $$p > 0$$ exists for $$F$$. Then there exists a function $$g(x)$$ defined on $$(0, 1)$$, depending only on $$p$$, and such that $$\lim_{s\to+0}g(s)=0$$ and $| F(x)-\Phi(x)| \leq\frac{\lambda_p+g(L)}{1+| x| ^p},\quad x\in\mathbb R,$ if $$0\leq L\leq 1$$.
Next, the authors deal with the generalized global version of the central limit theorem proved by R. P. Agnew [Proc. Natl. Acad. Sci. USA 40, 800–804 (1954; Zbl 0055.36703)].
Theorem C. Let $$\{F_n\}$$ be a sequence of distribution functions such that $$\int_{-\infty}^{+\infty}xdF_n(x)=0$$, $$\int_{-\infty}^{+\infty}x^2dF_n(x)=1$$. Let $$F_n\overset {w}{} \Phi$$ as $$n\to\infty$$. Then, for all $$r>1/2$$, $\int_{-\infty}^{+\infty}| F_n(x)-\Phi(x)| ^{r}\,dx\to0,\quad n\to\infty.$ The authors complete Agnew’s result by considering the case $$r = 1/2$$.
Theorem 3. Let $$\{F_n\}$$ be a sequence of distribution functions satisfying the conditions of the previous theorem. Then $\int_{-\infty}^{+\infty} \frac{| F_n(x)-\Phi(x)| ^{1/2}} {(\ln(1 + | x| ))^{1+\delta}}\,dx\to 0,\quad n\to\infty$ for all $$\delta>0$$.
A version of the results of G. Laube [Metrika 20, 103–105 (1973; Zbl 0257.60013)] and A. Rosalsky [Int. J. Math. Math. Sci. 11, No. 2, 365–374 (1988; Zbl 0645.60032)] on the global version of the central limit theorem is proved for the case $$r = 1/2$$.
Theorems which generalize the corresponding results by P. L. Hsu and H. Robbins [Proc. Natl. Acad. Sci. USA 33, 25–31 (1947; Zbl 0030.20101)] and C. C. Heyde [J. Appl. Probab. 12, 173–175 (1975; Zbl 0305.60008)] on the complete convergence of a sequence of random variables are proved.

### MSC:

 60J05 Discrete-time Markov processes on general state spaces

### Keywords:

Lévy distance; central limit theorem; complete convergence
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