Functional central limit theorems for self-normalized partial sums of linear processes

We prove an invariance principle under self-normalization by blocks for linear processes with summable filters and i.i.d. innovations in the domain of attraction of the normal distribution.     

Invariance principles for sums of Banach space valued random elements and empirical processes

Summary Almost sure and probability invariance principles are established for sums of independent not necessarily measurable random elements with values in a not necessarily separable Banach space. It is then shown that empirical processes readily fit into this general framework. Thus we bypass the problems of measurability and topology characteristic for the previous theory of weak convergence of empirical processes.Both authors were partially supported by NSF grants. This work was done while the second author was visiting the M.I.T. Mathematics Department     

A nonclassical law of the iterated logarithm for self-normalized partial sums

Let {X _n ,?n≧1} be a sequence of nondegenerate, symmetric, i.i.d. random variables which are in the domain of attraction of the normal?law?with zero means and possibly infinite variances. Denote ${S_{n}=\sum_{i=1}^{n} X_{i}}$ , ${V_{n}^{2}=\sum_{i=1}^{n} X_{i}^{2}}$ . Then we prove that there is a sequence of positive constants {b(n),?n≧1} which is defined by Klesov and Rosalsky [11], is monotonically approaching infinity and is not asymptotically equivalent to loglogn but is such that $\displaystyle \limsup_{n\to\infty} \frac{|S_n|}{\sqrt{2V_n^2b(n)}}= 1$ almost surely if some additional technical assumptions are imposed.     

Invariance principles for tempered fractionally integrated processes

We discuss invariance principles for autoregressive tempered fractionally integrated moving averages in $\alpha$-stable $(1<\alpha \le 2)$ i.i.d. innovations and related tempered linear processes with vanishing tempering parameter $\underset{N\to \infty }{lim}\lambda ∕N={\lambda }_1$. We show that the limit of the partial sums process takes a different form in the weakly tempered (${\lambda }_1=0$), strongly tempered (${\lambda }_1=\infty$), and moderately tempered ($0<{\lambda }_1<\infty$) cases. These results are used to derive the limit distribution of the ordinary least squares estimate of AR(1) unit root with weakly, strongly, and moderately tempered moving average errors.     

Invariance principles and Gaussian approximation for strictly stationary processes

We show that in any aperiodic and ergodic dynamical system there exists a square integrable process the partial sums of which can be closely approximated by the partial sums of Gaussian i.i.d. random variables. For both weak and strong invariance principles hold.

     

A strong approximation of self-normalized sums

Let {X,Xn,n1} be a sequence of independent identically distributed random variables with EX=0 and assume that EX2I(|X|≤x) is slowly varying as x→∞,i.e.,X is in the domain of attraction of the normal law.In this paper a Strassen-type strong approximation is established for self-normalized sums of such random variables.     

Estimation of distribution tails for normalized and self-normalized sums

A combinatorial inequality is derived. This inequality is applied to obtain new estimates for probabilities of large deviations of normalized and self-normalized sums of independent and dependent positive random values. As a consequence, an estimate from above is derived for the strong law of large numbers. Bibliography: 9 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 294, 2002, pp. 77–87.This research was supported in part by the Ministry of Education of Russia, grant E00-1.0-45, and by the Russian Foundation for Basic Research, grant 02-01-01099a.Translated by V. A. Egorov.     

On the interrelation of almost sure invariance principles for certain stochastic adaptive algorithms and for partial sums of random variables

Since the novel work of Berkes and Philipp⁽³⁾ much effort has been focused on establishing almost sure invariance principles of the form (1) $$\left| {\sum\limits_{i = 1}^{|\_t\_|} {x_1 - X_t } } \right| \ll t^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - \gamma } $$ where {x _i ,i=1,2,3,...} is a sequence of random vectors and {X _t ,t>-0} is a Brownian motion. In this note, we show that if {A _k ,k=1,2,3,...} and {b _k ,k=1,2,3,...} are processes satisfying almost-sure bounds analogous to Eq. (1), (where {X _t ,t≥0} could be a more general Gauss-Markov process) then {h _k ,k=1,2,3...}, the solution of the stochastic approximation or adaptive filtering algorithm (2) $$h_{k + 1} = h_k + \frac{1}{k}(b_k - A_k h_k )for{\text{ }}k{\text{ = 1,2,3}}...$$ also satisfies and almost sure invariance principle of the same type.     

Invariance principles for renewal processes when only moments of low order exist

Starting from recent strong and weak approximations to the partial sums of i.i.d. random vectors (cf. U. Einmahl, Ann. Probab., 15 1419–1440), some corresponding invariance principles are developed for associated renewal processes and random sums. Optimality of the approximation is proved in the case when only two moments exist. Among other applications, a Darling-Erdös type extreme value theorem for renewal processes will be derived.     

An almost sure central limit theorem for self-normalized weighted sums

Let X, X1 , X2 , ··· be a sequence of nondegenerate i.i.d. random variables with zero means, which is in the domain of attraction of the normal law. Let {a ni , 1≤i≤n, n≥1} be an array of real numbers with some suitable conditions. In this paper, we show that a central limit theorem for self-normalized weighted sums holds. We also deduce a version of ASCLT for self-normalized weighted sums.     

Invariance principles for Gaussian sequences

An almost sure invariance principle is proved for stationary Gaussian sequences whose covariances r(n) satisfy r(n) = O (n ^–1–)for some >0.     

A note on the normal approximation error for randomly weighted self-normalized sums

Let X = {X _n } _n≥1 and Y = {Y _n } _n≥1 be two independent random sequences. We obtain rates of convergence to the normal law of randomly weighted self-normalized sums $$\psi _n \left( {X,Y} \right) = \sum\nolimits_{i = 1}^n {{{X_i Y_i } \mathord{\left/ {\vphantom {{X_i Y_i } {V_n , V_n }}} \right. \kern-\nulldelimiterspace} {V_n , V_n }}} = \sqrt {Y_1^2 + \cdots + Y_n^2 } .$$ . These rates are seen to hold for the convergence of a number of important statistics, such as for instance Student’s t-statistic or the empirical correlation coefficient.     

Precise asymptotics in the deviation probability series of self-normalized sums

Let X,X₁,X₂,… be a sequence of nondegenerate i.i.d. random variables with zero means. Set Sn=X₁+?+Xn and . In the present paper we examine the precise asymptotic behavior for the general deviation probabilities of self-normalized sums, Sn/Wn. For positive functions g(x), ?(x), α(x) and κ(x), we obtain the precise asymptotics for the following deviation probabilities of self-normalized sums:

     

A weak invariance principle for self-normalized products of sums of mixing sequences

Let variables in the {X, Xn, n ≥ 1} be a sequence of strictly stationary φ-mixing positive random domain of attraction of the normal law. Under some suitable conditions the principle for self-normalized products of partial sums is obtained.