共查询到20条相似文献,搜索用时 31 毫秒
1.
Let Z = {Z0, Z1, Z2,…} be a martingale, with difference sequence X0 = Z0, Xi = Zi ? Zi ? 1, i ≥ 1. The principal purpose of this paper is to prove that the best constant in the inequality λP(supi |Xi| ≥ λ) ≤ C supiE |Zi|, for λ > 0, is C = (log 2)?1. If Z is finite of length n, it is proved that the best constant is . The analogous best constant Cn(z) when Z0 ≡ z is also determined. For these finite cases, examples of martingales attaining equality are constructed. The results follow from an explicit determination of the quantity Gn(z, E) = supzP(maxi=1,…,n |Xi| ≥ 1), the supremum being taken over all martingales Z with Z0 ≡ z and E|Zn| = E. The expression for Gn(z,E) is derived by induction, using methods from the theory of moments. 相似文献
2.
A real-valued adapted sequence of random variables is an amart if and only if it can be written as a sum of a martingale and a sequence dominated in absolute value by a Doob potential, i.e., a positive supermartingale that converges to 0 in L1. The same holds for vector-valued uniform amarts with the norm replacing the absolute value. 相似文献
3.
A procedure is given for proving strictness of some sharp, infinite-sequence martingale inequalities, which arise from sharp, finite-sequence martingale inequalities attained by degenerating extremal distributions. The procedure is applied to obtain strictness of the sharp inequalities of Cox and Kemperman and of Cox (sharp form of Burkholder's inequality) for all nontrivial martingale difference sequences X0,X1,…. 相似文献
4.
Let {Xn} be a stationary Gaussian sequence with E{X0} = 0, {X20} = 1 and E{X0Xn} = rnn Let cn = (2ln n), bn = cn? c-1n ln(4π ln n), and set Mn = max0 ?k?nXk. A classical result for independent normal random variables is that Berman has shown that (1) applies as well to dependent sequences provided rnlnn = o(1). Suppose now that {rn} is a convex correlation sequence satisfying rn = o(1), (rnlnn)-1 is monotone for large n and o(1). Then for all x, where Ф is the normal distribution function. While the normal can thus be viewed as a second natural limit distribution for {Mn}, there are others. In particular, the limit distribution is given below when rn is (sufficiently close to) γ/ln n. We further exhibit a collection of limit distributions which can arise when rn decays to zero in a nonsmooth manner. Continuous parameter Gaussian processes are also considered. A modified version of (1) has been given by Pickands for some continuous processes which possess sufficient asymptotic independence properties. Under a weaker form of asymptotic independence, we obtain a version of (2). 相似文献
5.
Let (Xn)n? be a sequence of real, independent, not necessarily identically distributed random variables (r.v.) with distribution functions FXn, and Sn = Σi=1nXi. The authors present limit theorems together with convergence rates for the normalized sums ?(n)Sn, where ?: → +, ?(n) → 0, n → ∞, towards appropriate limiting r.v. X, the convergence being taken in the weak (star) sense. Thus higher order estimates are given for the expression ∝f(x) d[F?(n)Sn(x) ? FX(x)] which depend upon the normalizing function ?, decomposability properties of X and smoothness properties of the function f under consideration. The general theorems of this unified approach subsume O- and o-higher order error estimates based upon assumptions on associated moments. These results are also extended to multi-dimensional random vectors. 相似文献
6.
Klaus D. Schmidt 《Journal of multivariate analysis》1980,10(1):123-134
For certain types of stochastic processes {Xn | n ∈ }, which are integrable and adapted to a nondecreasing sequence of σ-algebras n on a probability space (Ω, , P), several authors have studied the following problems: IfSdenotes the class of all stopping times for the stochastic basis {n | n ∈ }, when isfinite, and when is there a stopping time for which this supremum is attained? In the present paper we set the problem in a measure theoretic framework. This approach turns out to be fruitful since it reveals the root of the problem: It avoids the use of such notions as probability, null set, integral, and even σ-additivity. It thus allows a considerable generalization of known results, simplifies proofs, and opens the door to further research. 相似文献
7.
Bong Dae Choi 《Journal of multivariate analysis》1984,14(2):248-267
The regularity of trajectories of continuous parameter process (Xt)t∈R+ in terms of the convergence of sequence E(XTn) for monotone sequences (Tn) of stopping times is investigated. The following result for the discrete parameter case generalizes the convergence theorems for closed martingales: For an adapted sequence (Xn)1≤n≤∞ of integrable random variables, lim Xn exists and is equal to X∞ and (XT) is uniformly integrable over the set of all extended stopping times T, if and only if lim E(XTn) = E(X∞) for every increasing sequence (Tn) of extended simple stopping times converging to ∞. By applying these discrete parameter theorems, convergence theorems about continuous parameter processes are obtained. For example, it is shown that a progressive, optionally separable process (Xt)t∈R+ with E{XT} < ∞ for every bounded stopping time T is right continuous if lim E(XTn) = E(XT) for every bounded stopping time T and every descending sequence (Tn) of bounded stopping times converging to T. Also, Riesz decomposition of a hyperamart is obtained. 相似文献
8.
9.
A continuous-parameter ascending amart is a stochastic process (Xt)t
+ such that E[Xτn] converges for every ascending sequence (τn) of optional times taking finitely many values. A descending amart is a process (Xt)t
+ such that E[Xτn] converges for every descending sequence (τn), and an amart is a process which is both an ascending amart and a descending amart. Amarts include martingales and quasimartingales. The theory of continuous-parameter amarts parallels the theory of continuous-parameter martingales. For example, an amart has a modification every trajectory of which has right and left limits (in the ascending case, if it satisfies a mild boundedness condition). If an amart is right continuous in probability, then it has a modification every trajectory of which is right continuous. The Riesz and Doob-Meyer decomposition theorems are proved by applying the corresponding discrete-parameter decompositions. The Doob-Meyer decomposition theorem applies to general processes and generalizes the known Doob decompositions for continuous-parameter quasimartingales, submartingales, and supermartingales. A hyperamart is a process (Xt) such that E[Xτn] converges for any monotone sequence (τn) of bounded optional times, possibly not having finitely many values. Stronger limit theorems are available for hyperamarts. For example: A hyperamart (which satisfies mild regularity and boundedness conditions) is indistinguishable from a process all of whose trajectories have right and left limits. 相似文献
10.
Let X be a random vector with values in n and a Gaussian density f. Let Y be a random vector whose density can be factored as k · f, where k is a logarithmically concave function on n. We prove that the covariance matrix of X dominates the covariance matrix of Y by a positive semidefinite matrix. When k is the indicator function of a compact convex set A of positive measure the difference is positive definite. If A and X are both symmetric Var(a · X) is bounded above by an expression which is always strictly less than Var(a · X) for every a ∈ n. Finally some counterexamples are given to show that these results cannot be extended to the general case where f is any logarithmically concave density. 相似文献
11.
Philip L. Bowers 《Topology and its Applications》1984,17(2):173-187
It is shown that if G is an arbitrary upper semicontinuous decomposition of En for which π(NG embeds in Sm for some m?3, then the decomposition space embeds as a closed subset of En+m+1. The proof consists of constructing a cell-like upper semicontinuous decomposition G? of En+m+1 which intersects En to yield precisely G and using Edwards' Cell-Like Approximation Theorem to show that G? is shrinkable. As an immediate corollary, embeds in En+2k+2 whenever G is an arbitrary k-dimensional upper semicontinuous decomposition of En. This is an improvement of (n?1)-dimensions over the corresponding dimension theoretic result and examples due to Daverman show that this result is sharp in case n is odd and off by no more than one dimension in case n is even. 相似文献
12.
The following estimate of the pth derivative of a probability density function is examined: , where hk is the kth Hermite function and Σi = 1nhk(p)(Xi) is calculated from a sequence X1,…, Xn of independent random variables having the common unknown density. If the density has r derivatives the integrated square error converges to zero in the mean and almost completely as rapidly as O(n?α) and O(n?α log n), respectively, where . Rates for the uniform convergence both in the mean square and almost complete are also given. For any finite interval they are O(n?β) and , respectively, where . 相似文献
13.
Theorem. Let Xn, n ≥ 1, be a sequence of tight random elements taking values in a separable Banach space B such that |Xn|, n ≥ 1, is uniformly integrable. Let ank, n ≥ 1, k ≥ 1, be a double array of real numbers satisfying Σk ≥ 1 |ank| ≤ Γ for every n ≥ 1 for some positive constant Γ. Then Σk ≥ 1ankXk, n ≥ 1, converges to 0 in probability if and only if Σk ≥ 1ankf(Xk), n ≥ 1, converges to 0 in probability for every f in the dual space B1. 相似文献
14.
Robert M Burton 《Statistics & probability letters》1985,3(6):315-316
If a sequence of random variables Xn converges to X in probability we know little about the pointwise behavior Xn(ω). In this note we show that if Xn converges to X quickly enough (for example, like n?α for α > 0) then, for almost all ω, Xn(ω) converges to X(ω) outside a set of density zero. 相似文献
15.
Robert Chen 《Journal of multivariate analysis》1978,8(2):328-333
Let {Xn}n≥1 be a sequence of independent and identically distributed random variables. For each integer n ≥ 1 and positive constants r, t, and ?, let Sn = Σj=1nXj and . In this paper, we prove that (1) lim?→0+?α(r?1)E{N∞(r, t, ?)} = K(r, t) if E(X1) = 0, Var(X1) = 1, and E(| X1 |t) < ∞, where 2 ≤ t < 2r ≤ 2t, , and ; (2) if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E(|X1|t) < ∞, where G(t, ?) = E{N∞(t, t, ?)} = Σn=1∞nt?2P{| Sn | > ?n} → ∞ as ? → 0+ and , i.e., H(t, ?) goes to infinity much faster than G(t, ?) as ? → 0+ if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E(| X1 |t) < ∞. Our results provide us with a much better and deeper understanding of the tail probability of a distribution. 相似文献
16.
Marvin I. Knopp 《Journal of Number Theory》1980,12(1):2-9
If h, k ∈ Z, k > 0, the Dedekind sum is given by , with , . The Hecke operators Tn for the full modular group SL(2, Z) are applied to log η(τ) to derive the identities (n ∈ Z+) , where (h, k) = 1, k > 0 and σ(n) is the sum of the positive divisors of n. Petersson had earlier proved (1) under the additional assumption k ≡ 0, h ≡ 1 (mod n). Dedekind himself proved (1) when n is prime. 相似文献
17.
N.R. Mohan 《Stochastic Processes and their Applications》1976,4(2):157-165
Let X1, X2, … be a sequence of independent and identically distributed random variables with mean zero such that the common distribution function belongs to the domain of attraction of a stable law Gα,β with 1<α<2 and β=1 or α=2. If Sn=X1+…Xn and N(ξ)=min{k:Sk>ξ}, ξ>0, then it is shown that , 0<t<1, converges weakly under the Skorohod J1-topology to a stable subordinator of index , where B1(n) depends on the norming constant for Sn. 相似文献
18.
Joseph Gardiner 《Journal of multivariate analysis》1982,12(2):230-247
Let Xn,1 ≤ Xn,2 ≤ … ≤ Xn,n be the ordered variables corresponding to a random sample of size n with respect to a family of probability measures {Pθ:θ ∈ Θ} where Θ is an open subset of the real line. In many practical situations the Xn,i are the observables and experimentation must be curtailed prior to Xn,n. If τn is a stopping variable adapted to the σ-fields {σ(Xn,1,…,Xn,k): 1 ≤ k ≤ n} and Pn,θ the projection of Pθ onto σ(Xn,1,…,Xn,τn), the local asymptotic normality of the stopped progressively censored likelihood ratio statistics is established with θ, and θ, u held fixed, under certain conditions on the underlying distribution and on τn. Conditions are also given to ensure the local asymptotic normality of likelihood ratio statistics where the underlying observations are given in a series scheme. 相似文献
19.
Kamal C Chanda 《Statistics & probability letters》1985,3(5):261-268
Let {Xt; t = 1, 2,…} be a linear process with a location parameter θ defined by Xt ? θ = Σ0∞grZt?r where {Zt; t = 0, ±1,…} is a sequence of independent and identically distributed random variables, with E∥Z1∥δ < ∞ for some δ > 0. If δ ? 1 we assume further than E(Z1) = 0. Let η = δ if 0 < δ < 2, and η = 2 if δ ? 2. Then assume that Σ0∞∥ gr ∥η < ∞. Consider the class of estimators given by is of the form cnt = Σp = 0sβnptp for some s ? 0. An attempt has been made to investigate the distributional properties of in large samples for various choices of βnp (0 ? p ? s), s, and the distribution of Z1 under the constraints Σ0∞rkgr = 0, 0 ? k ? q where q in an arbitrary integer, 0 ? q ? s. 相似文献
20.
R.A. Maller 《Stochastic Processes and their Applications》1980,10(1):65-73
We show that if Xi is a stationary sequence for which converges to a finite non zero random variable of constant sign, where Sn=X1+X2+?+Xn and Bn is a sequence of constants, then Bn is regularly varying with index 1. If in addition is finite, then is finite, and if in addition to this Xi satisfies an asymptotic independence condition, . 相似文献