Partial Sums Sn Research Articles

Divergent fourier series: Numerical experiments

An open problem in the theory of Fourier series is whether there are functions f ∈ L1 such that the partial sums Sn(f, x) diverge faster than log log n, almost everywhere in x. For a class of particularly ‘bad’ functions Kahane proved that the rate of divergence is faster than o(log log n). We give here a probabilistic interpretation of the Kahane result, which shows that the record values of the sums Sn(f, x) should behave essentially as the record values of a sequence of independent identically distributed random variables, for which we deduce the divergence rate log log n. Numerical computation is in good agreement with the prediction. One can argue that the Kahane examples are in some sense ‘optimal’, and conclude that, under this assumption, ...(log log n) is the highest possible rate for divergence almost everywhere of the Fourier partial sums for L1 functions.

A useful generalization of renewal theory: counting processes governed by non-negative Markovian increments

Let N(t) be a counting process associated with a sequence of non-negative random variables (Xj)1∞ where the distribution of Xn+1 depends only on the value of the partial sum Sn = Σj=1nXj. In this paper, we study the structure of the function H(t) = E[N(t)], extending the ordinary renewal theory. It is shown under certain conditions that h(t) = (d/dt)H(t) exists and is a unique solution of an extended renewal equation. Furthermore, sufficient conditions are given under which h(t) is constant, monotone decreasing and monotone increasing. Asymptotic behavior of h(t) and H(t) as t → ∞ is also discussed. Several examples are given to illustrate the theoretical results and to demonstrate potential use of the study in applications.

A useful generalization of renewal theory: counting processes governed by non-negative Markovian increments

Let N(t) be a counting process associated with a sequence of non-negative random variables (X j)1 ∞ where the distribution of Xn +1 depends only on the value of the partial sum Sn = Σj=1 n Xj . In this paper, we study the structure of the function H(t) = E[N(t)], extending the ordinary renewal theory. It is shown under certain conditions that h(t) = (d/dt)H(t) exists and is a unique solution of an extended renewal equation. Furthermore, sufficient conditions are given under which h(t) is constant, monotone decreasing and monotone increasing. Asymptotic behavior of h(t) and H(t) as t → ∞ is also discussed. Several examples are given to illustrate the theoretical results and to demonstrate potential use of the study in applications.

4519621 Chuck for a machine tool : Norifum Okamoto, Fukui, Japan assigned to Okamoto Seiki Kogyo Kabushiki Kaisha

We consider partial sums Sn=X1+X2+⋯+Xn,n∈N, of i.i.d. random variables with moments E(X1)=0,E(X12)=σ2 and sup{t∈R:E(exp((t|X1|)1/psgn(X1))<∞}∈(0,∞) and show thatwith some explicit function ϕ(·). A related result for random variables with exponentially thin tails has recently been shown by Steinebach, extending a result given by Shao.

4519733 Method and apparatus for automatically exchanging a workpiece in a machine tool : Hans Gregg, Steinebach, Federal Republic Of Germany assigned to Carl Hurth Maschinenund Zahnradfabrik GmbH & Co

We consider partial sums Sn=X1+X2+⋯+Xn,n∈N, of i.i.d. random variables with moments E(X1)=0,E(X12)=σ2 and sup{t∈R:E(exp((t|X1|)1/psgn(X1))<∞}∈(0,∞) and show thatwith some explicit function ϕ(·). A related result for random variables with exponentially thin tails has recently been shown by Steinebach, extending a result given by Shao.

On an improved Erdő‐Rényi‐type law for increments of partial sums

AbstractLet X1, X2,… be an independently and identically distributed sequence with ξX1 = 0, ξ exp (tX1 &lt; ∞ (t ≧ 0) and partial sums Sn = X1 + … + Xn. Consider the maximum increment D1 (N, K) = max0≤n ≤ N ‐ K (Sn + K ‐ Sn)of the sequence (Sn) in (0, N) over a time K = KN, 1 ≦ K ≦ N. Under appropriate conditions on (KN) it is shown that in the case KN/log N → 0, but KN/(log N)1/2 → ∞, there exists a sequence (αN) such that K‐1/2 D1 (N, K) ‐ αN converges to 0 w. p. 1. This result provides a small increment analogue to the improved Erdős‐Rényi‐type laws stated by Csörgő and Steinebach (1981).

On the (J, pn, qn) method of summation

In the following discussion we shall assume that pn≧0, qn≧0 for all n and that qn + 1 &gt; qn → ∞. The (J, pn, qn) method of summation is defined as follows.The series with the partial sum sn, is called summable (J, pn, qn) to s, and we write if the seriesand converge to the sum functions p*(x) and p(s)(x) respectively for 0&lt;x&lt;1 and if τ(x) = p(s)(x)/p*(x)→s as x→1–0.

On two summability methods

Let Σan be a given infinite series with partial sums sn, and rn = nan. By and we denote the nth Cesáro means of order α (α –1) of the sequences (sn) and (rn), respectively. The series Σan is said to be absolutely summable (C, a) with index k, or simply summable |C, α|k, k ≥ 1, if

The asymptotic ruin problem when the healthy and sick periods form an alternating renewal process

Let {T1, Y1}∞i=1 be a sequence of positive independent random variables. Let, also, Z1 = βY′1 − πTi, i = 1, 2, …, where Y′1 = Max(0, Yi − w), w ⩾ 0, and where β < 0 and π is such that E(Z1) < 0. We consider the random walk of partial sums Sn = ϵni=1Zi in the presence of an absorbing region (u, ∞), u ⩾ 0, and S0 ≡ 0. Of interest is ψ(u) = Pr(S̄ ≤ u) where S̄ = Sup(0, S1, S2, …, Sn, …).

On the strong law of large numbers for dependent random variables

Let {Xn}n=1/∞ be a sequence of random variables with partial sumsSn, and let {ie241-1} be the σ-algebra generated byX1,…,Xn. Letf be a function fromR toR and suppose {ie241-2}. Under conditions off and moment conditions on theX'ns, we show thatSn/n converges a.e. (almost everywhere). We give several applications of this result.

Some properties of the Chebyshev method

Several properties of the Chebyshev method of summability, defined by G. G. Bilodeau, are investigated. Specifically, it is shown that the Chebyshev method is translative and is a Gronwall method. It is shown that the de Vallee Poussin method is stronger than the Chebyshev method, and that the Chebyshev method is not stronger than the (C, 1) method. The final result shows that the Chebyshev method exhibits the Gibbs phenomenon. Let Σ( — iyut be an alternating series with partial sums sn = ΣΓ=o( —l)*w< Define a sequence of polynomials {PJt)} by PJt) = Σ*=o«•***, P*(l) = 1, n = 0, 1, 2, •••. The series £(-1)%, will be called summable (PJ to the value s if lim σ(Pn) = 8, where σ(Pn) — ΣLo #«&£*• Bilodeau [1] considered the following question. What are sufficient conditions on Pn for σ(Pn) to speed up the rate of convergence of a convergent sequence {sn}Ί For sequences {un} which are moment sequences, i.e., un has the representation un = \tnda(t), Jo where a(t)eBV[0fϊ\, he obtains the estimate \σ(Pn) — s\/\rn ^

Growth of partial sums of divergent series

Let Σ f ( n ) \Sigma f(n) be a divergent series of decreasing positive terms, with partial sums s n {s_n} , where f decreases sufficiently smoothly; let φ ( x ) = ∫ 1 x f ( t ) d t \varphi (x) = \smallint _1^xf(t)dt and let ψ \psi be the inverse of φ \varphi . Let n A {n_A} be the smallest integer n such that s n ⩾ A {s_n} \geqslant A but s n − 1 &gt; A ( A = 2 , 3 , … ) {s_{n - 1}} &gt; A(A = 2,3, \ldots ) ; let γ = lim { Σ 1 n f ( k ) − φ ( n ) } \gamma = \lim \{ \Sigma _1^nf(k) - \varphi (n)\} be the analog of Euler’s constant; let m = [ ψ ( A − γ ) ] m = [\psi (A - \gamma )] . Call ω \omega a Comtet function for Σ f ( n ) \Sigma f(n) if n A = m {n_A} = m when the fractional part of ψ ( A − γ ) \psi (A - \gamma ) is less than ω ( A ) \omega (A) and n A = m + 1 {n_A} = m + 1 when the fractional part of ψ ( A − γ ) \psi (A - \gamma ) is greater than ω ( A ) \omega (A) . It has been conjectured that ω ( A ) = 1 / 2 \omega (A) = 1/2 is a Comtet function for Σ 1 / n \Sigma 1/n . It is shown that in general there is a Comtet function of the form \[ ω ( A ) = 1 2 + 1 24 { | f ′ ( m ) | / f ( m ) } ( 1 + o ( 1 ) ) . \omega (A) = \frac {1}{2} + \frac {1}{24} \left \{ |f\prime (m)|/f(m) \right \} (1 + o(1)). \] For Σ 1 / n \Sigma 1/n there is a Comtet function of the form 1 / 2 + 1 / ( 24 ) { 1 / ( 48 m 2 ) } ( 1 + o ( 1 ) ) 1/2 + 1/(24) \left \{ 1/(48m^2) \right \} (1 + o(1)) . Some numerical results are presented.

The asymptotic distribution of the range and other functions of partial sums of stationary processes

Let ℰn, n = 1, 2, ⋯ , be the net input in a reservoir during the nth period of time, and set S0 = 0, Sn = ℰ1 + … + ℰ n, = 1, 2, ⋯ . Many quantities of interest, such as range, first‐passage times, and duration of deficit period, are functions of the partial sums Sn. In this paper it is pointed out that the functional central limit theorem, which has been previously used to obtain asymptotic results for independent and identically distributed ℰn, can be applied to a class of stationary sequences as well. To this class belong m‐dependent, Markov, autoregressive, and autoregressive‐moving average types of stationary processes.

Convergence in the mean of the Fourier series in orthogonal polynomials

For weights p(t) and q(t) with a finite number of power-law-type singularities we obtain necessary and sufficient conditions for the inequality $$\parallel s_n^{(p)} (f)q\parallel _{L^\eta - (1,1)} \leqslant C\parallel fq\parallel _{L^\eta ( - 1,1)'}$$ to hold, where sn(p)(f) is a partial sum of the Fourier series of the function f in terms of polynomials orthogonal on [−1, 1] with weight p(t). This inequality is used to solve the problem concerning convergence in the mean and also convergence almost everywhere of the partial sums sn(p)(f).

Weak laws for dependent sums

A general weak law of large numbers for sums Sn-=X1+*+X,n is proved. That is, without assuming the existence of any moments, and allowing any sort of dependence structure, conditions are given for S,jn-O in probability; the conditions are not necessary. However they are sufficient for a much stronger statement, namely that S, Iv, >0 in probability in many cases where positive, integer-valued random variables v,,oo. Introduction. Let {Xi} be a sequence of random variables on a probability space (Q, d, p) with partial sums Sn,=X1+ +X,. For reals 0 0 are sufficiently well-behaved, then also Sv= (Vn), a weak law for random sums; the v,, need not be independent of the Xn. This work began with the proof of the corollary and depended on the truth of Proposition 1. However the obvious sources ([2] and [4]) made no mention of the weak law in the present context, a surprising fact that necessitated proving this simple result. The results. For each n> 1 let -_ Cs 'n+c be a sigma field of subsets of Q containing the Borel field B(Sn) generated by Sn and put Received by the editors March 19, 1973. AMS (MOS) subject classifications (1970). Primary 60F05, 60F99, 60G45.

The Erdős-Rényi new law of large numbers for weighted sums

From the partial sums S n {S_n} of the first N N random variables of a sequence of independent, identically distributed random variables, N − K + 1 N - K + 1 averages of the form K − 1 ( S n + K − S n ) {K^{ - 1}}({S_{n + K}} - {S_n}) can be constructed, one such average for each n n between 0 and N − K N - K , inclusive. If we denote by Σ ( N , K ) \Sigma (N,K) the maximum of those N − K + 1 N - K + 1 averages, then for a wide range of numbers λ \lambda , Erdös and Rényi (1970) proved that, as N → ∞ , Σ ( N , K ) → λ N \to \infty ,\Sigma (N,K) \to \lambda a.e. for K = [ C ( λ ) log ⁡ N ] K = [C(\lambda )\log N] , where C ( λ ) C(\lambda ) is a constant depending only on λ \lambda , not on N N . The objective of the present article is to extend the Erdös-Rényi theorem to the case of weighted sums. The main theorem bears a relation to the law of large numbers for weighted sums of Jamison, Orey, and Pruitt (1965) similar to the one borne by the Erdös-Rényi theorem to the ordinary strong law of large numbers.

On (J, pn)-summability of Fourier series

Let pn&gt;0 be such that pn diverges, and the radius of convergence of the power seriesis 1. Given any series σan with partial sums sn, we shall use the notationand

A Note on the Arc-Sine Law and Markov Random Sets

0. Introduction. The purpose of this note is to point out an extension of a theorem of Dynkin [2], [3, page 447] concerning processes to the context of Markov random sets (or, as we shall call them, semilinear Markov processes). Dynkin's result states that, if X1, X2, is a renewal sequence, i.e. a sequence of nonnegative, independent, identically distributed random variables, with partial sums Sn, and if x, is defined by

An Integral Over Function Space

Real functionsmay be identified with elements x = (x0, x1, x2, … ) of the sequence space l1 Since the unit sphere S∞ of l1 is compact under the weak* topology = topology of co-ordinatewise convergence, a countably additive measure on S∞ is induced by a positive linear functional E (integral) on C(S∞), the weak* continuous real-valued functions on S∞. There exists a natural integral over S∞ reducing towhen f is a function of x0 alone. The partial sums Sn = Sn(x) of the power series for x(t) then form a martingale and zero-or-one phenomena appear. In particular, if R(x) is the radius of convergence of the series and e is the base of the natural logarithms, it turns out that R(x) = e for almost all x in S∞. Applications of the integral to the theory of numerical integration, the original motivation, will appear in a later paper.

On interval recurrent sums of independent random variables

where Va(x) is the distribution function of the symmetric stable law of order a (with the characteristic function exp { I X } ). (B) The distribution for each Xi, i = 1, 2, * * *, is determined by a probability density f(x), such that f(x) LLP(oc, oc) for some p>1. Under the above conditions' it was shown by Kallianpur and Robbins [1] that the sequence of partial sums Sn is interval recurrent. This means that