Sum Of Processes Research Articles

Embedding the Finite Sampling Process at a Rate

A huge body of if and only if theorems can be obtained based on certain strong embedding theorems for the partial sum process $\mathbb{S}_n$ and the uniform empirical and quantile processes $\mathbb{U}_n$ and $\mathbb{V}_n$. This embedding was accomplished in 1986 by M. Csorgo, S. Csorgo, L. Horvath and D. Mason. Their embedding is beautifully formulated so that many necessary and sufficient type results can be established using it. It is worthwhile to have an accessible proof. Indeed, these authors have since produced two papers that obtain the essential form of their result without appealing to earlier work of J. Komlos, P. Major and G. Tusnady. Indeed, this present paper does this for the finite sampling process $\mathbb{R}_n$ and the weighted empirical process $\mathbb{W}_n$. These latter results are entirely new and are the main objectives of the present paper. The applications of these latter results will appear elsewhere.

A simple proof of a rate of embedding for the partial sum process

A simple proof of Skorokhod embedding of the partial sum process at a rate is given. Nothing more sophisticated than the HBjek‐RBnyi inequality, the monotone inequality, the one sided reflection principal for Brownian motion and Mill's ratio is used.

Determination of host-ion spin-lattice relaxation times from Gd 3+ EPR linewidths in NH 4R(SO 4) 2·4H 2O (P  Pr, Sm, Ce, Nd) single crystals

Spin-lattice relaxation times (τ) for the host rare-earth ions R 3+ (R  Sm, Ce, Nd) in NH 4R(SO 4) 2·4H 2O crystal lattices have been estimated at room-temperature from X-band EPR linewidths of Gd 3+ impurity ion, using an appropriate expression based on the second moment for crystals consisting of two different kinds of paramagnetic spins. In addition, the τ values for Pr 3+ ions in NH 4Pr(SO 4) 2·4H 2O lattice have been estimated in the 266–410 K temperature range from EPR linewidths of the impurity ion Gd 3+. The values of τ for Pr 3+ indicate that τ −1 varies as T 7 ( T = temperature) in the 266–296 K range, and as T 2 in the range 296–410 K. It is concluded that the T 7 behaviour is predominantly due to the sum process, while the T 2 behaviour is predominantly due to the Raman process.

Error Estimates in Generalized Trigonometric Hölder-Zygmund Norms

We consider Hölder-Zygmund spaces of 2\pi -periodic functions f: \mathbb R \to \mathbb C , where the k -th difference with step-size h of the r -th derivative in the L^p - or C -norm is bounded by a modulus-type function \omega (h) . For the Fourier sum and related approximation processes we investigate error estimates in corresponding Hölder-Zygmund norms if the smoothness of f is given by other Hölder-Zygmund conditions. The convergence order for the general case can be formulated in a simple manner. This allows us to state also Jackson-type theorems for such Banach spaces. Moreover, we give explicit values for the constants appearing in these estimates.

Bahadur-Kiefer-Type Processes

We establish strong and weak laws for Bahadur-Kiefer-type processes of the form $e_n + i_n$, where $i_n$ denotes the inverse of $e_n$. In particular, we provide a proof for the strong version of Theorem 1A of Kiefer (1970), together with similar results for renewal and partial sum processes.

Residual partial sum limit process for regression models with applications to detecting parameter changes at unknown times

Limit processes for sequences of stochastic processes defined by partial sums of linear functions of regression residuals are derived. They are Gaussian and are functions of standard Brownian motion. Cramér–von Mises type functionals defined on the partial sum processes are shown to converge in distribution to the same functionals defined on the limit processes. This result is then applied to derive the asymptotic forms of two-sided change detection statistics for linear regression models. These are derived for a variety of weight sequences and are shown to involve sums of Cramér–von Mises type stochastic integrals. Finally a methodology is developed to derive distributions of these stochastic integrals for the case of harmonic regression. This methodology is applicable to more general situations.

Almost sure and weak invariance principles for random variables attracted by a stable law

Let (Xn) be i.i.d. random variables belonging to the domain of normal attraction of a symmetric stable law with parameter 0<p<2. We study the a.s. and weak approximation of the partial sum process\(S(t) - \sum\limits_{n \leqq t} {X_n (t \geqq } 0)\) by a symmetric stable processGp(t). Stout proved an upper bound for the optimal remainder term in this approximation; we prove here a lower bound, leaving only a small gap between the upper and lower estimates. We also give a new method to obtain upper bounds. Finally, we prove analogues of these results in the case when a.s. approximation is replaced by approximation in probability.

Laws of the iterated logarithm for partial sum processes indexed by functions

We establish a bounded and a compact law of the iterated logarithm for partial sum processes indexed by classes of functions. We assume a growth condition on the metric entropy under bracketing. Examples show that our results are sharp. As a corollary we obtain new results for weighted sums of independent identically distributed random variables.

Lilly lecture 1988. Glucose counterregulation and its impact on diabetes mellitus.

Glucose counterregulation is the sum of processes that protect against development of hypoglycemia and that restore euglycemia if hypoglycemia should occur. In order of importance, the key counterregulatory factors are glucagon, epinephrine, growth hormone, cortisol, and hepatic autoregulation. These act primarily by increasing hepatic glucose output, initially via breakdown of glycogen and later by gluconeogenesis. In people without diabetes and in people with type II (non-insulin-dependent) diabetes, suppression of endogenous insulin secretion during hypoglycemia is also important in permitting full expression of the effects of counterregulation. People with diabetes are more prone to develop hypoglycemia for various reasons (e.g., insulin overdose, skipped meals, and intensive exercise); one that has recently been identified is impaired glucose counterregulation: patients with type I (insulin-dependent) diabetes (and to a lesser extent, patients with type II diabetes) lose the glucagon response to hypoglycemia; subsequent development of autonomic neuropathy with concomitant loss of the epinephrine response leads to almost complete paralysis of counterregulation and loss of recognition of hypoglycemia. To make matters worse, an episode of hypoglycemia that causes activation of counterregulation can lead to rebound hyperglycemia (Somogyi phenomenon); if this is improperly treated, brittle diabetes may follow. Thus, abnormalities in glucose counterregulation may predispose to severe hypoglycemia and prevent achievement of optimal glycemic control in patients with diabetes.

Log-fractional stable processes

The first problem attacked in this paper is answering the question whether all 1/α-self-similar α-stable processes with stationary increments are α-stable motions. The answer is yes for α = 2, no for 1⩽α<2 and unknown for 0<α<1. We single out the log-fractional stable processes for 1<α⩽2, different from α-stable motions for α≠2. They can be regarded as the limit of fractional stable processes as the exponent in the kernel tends to 0. The paper ends with a limit theorem for partial sum processes of moving averages of iid random variables in the domain of attraction of a strictly stable law, with log-fractional stable processes as limits in law. The conditions involve de Haan's class Π of slowly varying functions.

Limit laws for the modulus of continuity of the partial sum process and for the shepp statistic

Let S n denote the partial sum of an i.i.d. sequence of centred random variables having a finite moment generating function ø in a neighbourhood of zero. In this paper, we establish strong and weak limit laws for W n= max 1⩽i⩽n−k max 1⩽j⩽k (S i+j−S i), V n= max 1⩽i⩽n−k min 1⩽j⩽k (k⧸j)(S i+j−S i) and T n= max 1⩽i⩽n−k (S i+k(i)−S i , where 1⩽ k= k( n)⩽ n is an integer sequence that k( n)⧸ n→ 0 and lim inf n → ∞ k( n)⧸log n>0. Our results extend those of Deheuvels, Devroye and Lynch (1986), Deheuvels and Devroye (1987), Deheuvels and Steinebach (1987) and M.Csörgõ and Steinebach (1981).

Second-order approximations for certain stopped sums in extended renewal theory

Let (X0, Y0), (X1, Y1), · ·· be a sequence of independent two-dimensional random vectors such that (X1, Y1), (X2, Y2), · ·· are i.i.d. Let {(Sn, Un)}n≧0 be the associated sum process, and define for t ≧ 0 Under suitable conditions on (X0, Y0) and (X1, Y1) we derive expansions up to vanishing terms, as t→∞, for EUT(t), Var UT(t) and Cov (UT(t), T(t)). Corresponding results will be obtained for EUN(t), Var UN(t) and Cov (UN(t), N(t)) when X0, Χ1 are both almost surely non-negative and

Second-order approximations for certain stopped sums in extended renewal theory

Let (X0, Y0 ), (X1, Y 1), · ·· be a sequence of independent two-dimensional random vectors such that (X 1, Y 1), (X 2 , Y2 ), · ·· are i.i.d. Let {(Sn , Un )} n≧0 be the associated sum process, and define for t ≧ 0 Under suitable conditions on (X0 , Y 0) and (X1, Y 1) we derive expansions up to vanishing terms, as t→∞, for EUT(t) , Var UT(t) and Cov (UT(t) , T(t)). Corresponding results will be obtained for EUN(t) , Var UN (t) and Cov (UN (t) , N(t)) when X0, Χ 1 are both almost surely non-negative and

Some remarks on regression with autoregressive errors and their residual processes

We consider some linear regression models Y = Σα lfl(z) + X, where X is an autoregressive (AR) process. The residuals estimate the i.i.d. innovations sequence which drives the AR process. We then consider the partial sum process of the residuals and show they converge to Brownian bridges in certain cases. Some remarks are also made on similar processes when differencing is first applied to remove trends. When an AR process is differenced the residual partial sum can be asymptotically a random polynomial.

Some remarks on regression with autoregressive errors and their residual processes

We consider some linear regression models Y = Σα lfl (z) + X, where X is an autoregressive (AR) process. The residuals estimate the i.i.d. innovations sequence which drives the AR process. We then consider the partial sum process of the residuals and show they converge to Brownian bridges in certain cases. Some remarks are also made on similar processes when differencing is first applied to remove trends. When an AR process is differenced the residual partial sum can be asymptotically a random polynomial.

A central limit theorem for D( A)-valued processes

Let D ( A ) be the space of set-indexed functions that are outer continuous with inner limits, a generalization of D[0, 1]. This paper proves a central limit theorem for triangular arrays of independent D ( A ) valued random variables. The limit processes are not restricted to be Gaussian, but can be quite general infinitely divisible processes. Applications of the theorem include construction of set-indexed Lévy processes and a unified central limit theorem for partial sum processes and generalized empirical processes. Results obtained are new even for the D[0, 1] case.

The Law of Large Numbers for Partial Sum Processes Indexed by Sets

This note provides necessary and sufficient conditions for the law of large numbers for partial sum processes indexed by sets of $\lbrack 0, 1\rbrack^d$ and based on i.i.d. integrable random variables.

A strong law for a set-indexed partial sum process with applications to exchangeable and stationary sequences

Consider a set-indexed partial sum process { Σ jεJ n Y j , n ⩾ 1} where { Y n , n ⩾ 1} are identically distributed random variables and { J n , n ⩾ 1} is a nondecreasing sequence of finite sets of positive integers with j( n) = Card( J n ) → ∞. A strong law of the form Σ jεJ n Y j / b j( n) → 0 almost certainly is established where { b n , n ⩾ 1} are constants with b n / n → ∞. As special cases, new results are obtained for exchangeable and stationary sequences. The result for stationary sequences strengthens a weak law proved by Maller [9] in that the convergence is shown to be almost certain.

Invariance principles for partial sum processes and empirical processes indexed by sets

Let |xj,j∈ℕq} be independent identically formed random elements with values in a Banach spaceB, indexed byq-tuples of positive integers. We obtain the almost sure approximation as well as the approximation in probability of the partial sum process\(\{ \sum\limits_{j \in nA} {x_j ,A \in A\} } \) by a partial sum process\(\{ \sum\limits_{j \in nA} {y_j ,A \in A\} } \) asn→∞ uniformly over all setsA in a certain class\(A\) of subsets of theq-dimensional unit cube. Here |yj,j∈ℕq} are independent identically distributed Gaussian random variables. These results are then applied to obtain the approximation of empirical processes over sets and indexed by sets by Gaussian partial sum processes indexed by sets.

THE SUM OF FINITE MOVING AVERAGE PROCESSES

Abstract. It is well known that the sum of moving average processes is itself a moving average process. Existing theory does not provide formulae relating the innovations in the sum process to those in the component processes when some zeros of the autocovari‐ance function of the sum process are on the unit circle. This gap is filled here by first showing that these zeros are necessarily shared by the component processes. Conditions for the innovations in the sum process to be current are given. Throughout the paper contemporary innovations in the component processes are allowed to be correlated, replacing the usual assumption that they are uncorrelated.