A functional central limit theorem for regression models

A property of partial sums of regression least squares residuals and its applications

We consider the process of partial sums of moving averages of finite order with a regular varying memory function, constructed from a stationary sequence, variance of the sum of which is a regularly varying function. We study the Gaussian approximation of this process of partial sums with the aid of a certain class of Gaussian processes, and obtain sufficient conditions for the $C$-convergence in the invariance principle in the Donsker form

Recently, some researchers determined lower bounds for the normalized version of some special functions to its sequence of partial sums, e.g., Struve and Dini functions, Wright functions and Miller–Ross functions. In this paper, we determine lower bounds for the normalized Le Roy-type Mittag-Leffler function Fα,βγ(z)=z+∑n=1∞Anzn+1, where An=ΓβΓα(n−1)+βγ and its sequence of partial sums (Fα,βγ(z))m(z)=z+∑n=1mAnzn+1. Several examples of the main results are also considered.

Limit processes are obtained for the sequences of partial sums of polynomial regression residuals. Properties of linear and quadratic functionals on the sequences are discussed. Distribution theory for Cramer-von Mises type functionals is obtained. An indication is given of the relevance of these results to the problem of testing for change of regression at unknown times.

Let X,X 1,X 2,… be a sequence of non-degenerate i.i.d. random variables with mean zero. The best possible weighted approximations are investigated in D[0, 1] for the partial sum processes {S [nt], 0 ≦ t ≦ 1} where S n = Σ n j=1 X j , under the assumption that X belongs to the domain of attraction of the normal law. The conclusions then are used to establish similar results for the sequence of self-normalized partial sum processes {S [nt]=V n , 0 ≦ t ≦ 1}, where V 2 n = Σ n j=1 X 2 j . L p approximations of self-normalized partial sum processes are also discussed.

Let ϕ(z) be a fixed harmonic functions of the form $\varphi (z) = z + \sum\nolimits_{k = 2}^\infty {c_k z^k + } \overline {\sum\nolimits_{k = 1}^\infty {d_k z^k } } $ (d k ≥ c k ≥ c 2 > 0; k ≥ 2) and S H (c k , d k , δ) be the subclass of harmonic univalent functions f(z) which satisfy the inequality $\sum\nolimits_{k = 2}^\infty {c_k |a_k | + } \sum\nolimits_{k = 1}^\infty {d_k |b_k | \leqslant \delta } $ . In this paper, we establish some interesting results on the ratio of harmonic univalent functions to its sequences of partial sums. Relevant connections of the results presented here with various known results are briefly indicated.

Let α be a non-negative real number, λ≡{λ,n}(n≥0) a strictly increasing unbounded sequence with λ0≥0 and let be an arbitrary series with partial sums s≡{sn}. Writewhere s(t)=sn for λn<t≤λn+1, s(t)=0 for 0≤t≤λ0. The series ∑ an or the sequence of partial sums s={sn} is summable to ṡ by the Riesz method (R, λ, α) ifas ω→∞.

In the author's paper [7] it was proved that the fractional parts of the partial sums of an infinite series (of real terms) diverging to +∞ or −∞, in which the general term tends to zero, are everywhere dense in the closed unit interval. This result was extended to series of infinite oscillation (see Remark 4.1 of the said paper) on the argument that a sequence of partial sums having infinite oscillation has a subsequence that diverges to +∞ or −∞.

A random walk with counterbalanced steps is a process of partial sums \check S(n)=\check X_{1}+ \cdots + \check X_{n} whose steps \check X_{n} are given recursively as follows. For each n\geq 2 , with a fixed probability p , \check X_{n} is a new independent sample from some fixed law \mu , and with complementary probability 1-p , \check X_{n}= -\check X_{v(n)} counterbalances a previous step, with v(n) a uniform random pick from \{1, \ldots, n-\nobreak 1\} . We determine the asymptotic behavior of \check S(n) in terms of p and the first two moments of \mu . Our approach relies on a coupling with a reinforcement algorithm due to H. A. Simon, and on properties of random recursive trees and Eulerian numbers, which may be of independent interest. The method can be adapted to the situation where the step distribution \mu belongs to the domain of attraction of a stable law.

Empirical and partial sum processes with sample path in Banach function spaces

The main purpose of this paper is to draw attention to a simple and useful general "continuity theorem" type result, from which a great deal of limit theorems follow as almost immediate consequences. As an example, we give a new very short and transparent proof of the recent result by H. Teicher and C. Hagwood (A multidimensional CLT for maxima of normed sums); in fact, a much more general assertion is proved here. Another application of the main result establishes a correspondence between the convergence of empirical and quantile processes. (A similar result holds for the renewal and partial sums processes.)

A general class of fluctuation tests for parameter instability in an M‐estimation framework is suggested. Tests from this framework can be constructed by first choosing an appropriate estimation technique, deriving a partial sum process of the estimation scores that captures instabilities over time, and aggregating this process to a test statistic by using a suitable scalar functional. Inference for these tests is based on functional central limit theorems, which are derived under the null hypothesis of parameter stability and local alternatives. For (generalized) linear regression models, concrete tests are derived, which cover several known tests for (approximately) normal data but also allow for testing for parameter instability in regressions with binary or count data. The usefulness of the test procedures—complemented by powerful visualizations derived from these—is illustrated using Dow Jones industrial average stock returns, youth homicides in Boston, USA, and illegitimate births in Grossarl, Austria.

We discuss contemporaneous aggregation of independent copies of a triangular array of random‐coefficient processes with i.i.d. innovations belonging to the domain of attraction of an infinitely divisible law W. The limiting aggregated process is shown to exist, under general assumptions on W and the mixing distribution, and is represented as a mixed infinitely divisible moving average in (4). Partial sums process of is discussed under the assumption EW2 < ∞ and a mixing density regularly varying at the ‘unit root’ x = 1 with exponent β > 0. We show that the previous partial sums process may exhibit four different limit behaviors depending on β and the Lévy triplet of W. Finally, we study the disaggregation problem for in spirit of Leipus et al. (2006) and obtain the weak consistency of the corresponding estimator of ϕ(x) in a suitable L2 space.

Let s1, …, sn be generated governed by an r-state irreducible aperiodic Markov chain. The partial sum process is determined by a realization of states with s0 = α and the real-valued i.i.d. bounded variables Xαß associated with the transitions si = α, si+1 = β. Assume Χ αβ has negative stationary mean. The explicit limit distribution of the maximal segmental sum is derived. Computational methods with potential applications to the analysis of random Markov-dependent letter sequences (e.g. DNA and protein sequences) are presented.

We extend classical results by A. V. Nagaev [Izv. Akad. Nauk UzSSR Ser. Fiz.--Mat. Nauk 6 (1969) 17--22, Theory Probab. Appl. 14 (1969) 51--64, 193--208] on large deviations for sums of i.i.d. regularly varying random variables to partial sum processes of i.i.d. regularly varying vectors. The results are stated in terms of a heavy-tailed large deviation principle on the space of c\`{a}dl\`{a}g functions. We illustrate how these results can be applied to functionals of the partial sum process, including ruin probabilities for multivariate random walks and long strange segments. These results make precise the idea of heavy-tailed large deviation heuristics: in an asymptotic sense, only the largest step contributes to the extremal behavior of a multivariate random walk.