Analytic theory of linear forms of independent random variables

It is shown that in some characterizations of the Gaussian distribution, through natural properties of linear forms of independent, possibly equidistributed random variables, instead of the -algebra generated by one of the forms one can restrict oneself to the finite-dimensional space of polynomials of this form. Thus, the corresponding characterization theorems are significantly refined.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 153, pp. 37–44, 1986.     

A local limit theorem for independent random variables

A local limit theorem is given for independent noninteger random variables under a condition which is more general than one previously given, and which reduces, in the case of identically distributed random variables, to a well-known result.     

A limit theorem for the moments of sums of independent random variables

Forn≧1, letS _n=ΣX _n,i (1≦i≦r _n <∞), where the summands ofS _n are independent random variables having medians bounded in absolute value by a finite number which is independent ofn. Letf be a nonnegative function on (− ∞, ∞) which vanishes and is continuous at the origin, and which satisfies, for some for allt≧1 and all values ofx. Theorem.For centering constants c _n,let S _n − c _n converge in distribution to a random variable S. (A)In order that Ef(S_n − c_n) converge to a limit L, it is necessary and sufficient that there exist a common limit (B)If L exists, then L<∞ if and only if R<∞, and when L is finite, L=Ef(S)+R. Applications are given to infinite series of independent random variables, and to normed sums of independent, identically distributed random variables.     

Limit theorem for the maximum of gaussian independent random variables in the spaceC

We generalize the well-known asymptotic equality for the maximum of real Gaussian random variables to the case of random variables with values in the spaceC.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 7, pp. 1006–1008, July, 1995.     

An almost sure limit theorem for random sums of independent random variables in the domain of attraction of a semistable law

In this paper we prove an almost sure limit theorem for random sums of independent random variables in the domain of attraction of a p-semistable law and describe the limit law.     

A limit theorem for sums of independent,nonidentically distributed random variables

A generalization and refinement of Chen's theorem related to a strong law of large numbers for sums of independent, nonidentically distributed random variables are obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta in im. V. A. Steklova AN SSSR, Vol. 85, pp. 188–192, 1979.     

A limit theorem for multiple sums of identically distributed independent random variables

One investigates the asymptotic behavior (with respect to a small parameter) of series of probabilities of large deviations for sums of random variables with multidimensional indices.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 265–277, 1986.     

A bound on the Wasserstein-2 distance between linear combinations of independent random variables

We provide a bound on a distance between finitely supported elements and general elements of the unit sphere of ${?}^2\left({\mathbb{N}}^1\right)$. We use this bound to estimate the Wasserstein-2 distance between random variables represented by linear combinations of independent random variables. Our results are expressed in terms of a discrepancy measure related to Nourdin–Peccati’s Malliavin–Stein method. The main application is towards the computation of quantitative rates of convergence to elements of the second Wiener chaos. In particular, we explicit these rates for non-central asymptotic of sequences of quadratic forms and the behavior of the generalized Rosenblatt process at extreme critical exponent.     

On the rate of convergence in the global central limit theorem for random sums of independent random variables

We present upper bounds of the integral \( {\int}_{-\infty}^{\infty }{\left|x\right|}^l\left|\mathbf{P}\left\{{Z}_N<x\right\}-\varPhi (x)\right|\mathrm{d}x \) for 0 ≤ l ≤ 1 + δ, where 0 < δ ≤ 1, Φ(x) is a standard normal distribution function, and Z _N = \( {S}_N/\sqrt{\mathbf{V}{S}_N} \) is the normalized random sum with variance V S _N > 0 (S _N = X ₁ + · · · + X _N ) of centered independent random variables X ₁ ,X ₂ , . . . . The number of summands N is a nonnegative integer-valued random variable independent of X ₁ ,X ₂ , . . . .     

A limit theorem for a certain transform of sums of independent random variables

Summary A certain transformation of distribution functions (d.f.'s) of positive random variables (r.v.'s) has been studied by the author and Harkness in [2] and [3]. In this paper, a limit theorem concerning such a transform of convolutions of d.f.'s is proved.     

An analog of the Baum-Katz theorem for weakly dependent random variables

In this paper we study the limiting behavior of sums of dependent random variables under a strong mixing condition. We obtain conditions for which an analog of the Baum-Katz theorem holds and cite an example showing their optimality. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 360–368, March, 2000.     

A Glivenko-Cantelli theorem for empirical measures of independent but non-identically distributed random variables

The bounded-dual-Lipschitz and Prohorov distances from the ‘empirical measure’ to the ‘average measure’ of independent random variables converges to zero almost surely if the sequence of average measures is tight. Three examples are also given.