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1.
This paper discusses the asymptotic behaviors of the longest run on a countable state Markov chain.Let {Xa} a∈Z + be a stationary strongly ergodic reversible Markov chain on countablestate space S = {1,2,...}.Let TS be an arbitrary finite subset of S.Denote by Ln the length of the longest run of consecutive i's for i∈T,that occurs in the sequence X1,...,Xn.In this paper,we obtain a limit law and a week version of an Erds-Rényi type law for Ln.A large deviation result of Ln is also discussed.  相似文献   

2.
Let {Xn } be a Markov chain with transition probability pij = aj-(i-1)+ , i, j ≥ 0, where a j = 0 provided j 0, a 0 0, a 0 + a 1 1 and ∑∞ n=0 an = 1. Let μ =∑∞ n=1 nan . It is known that {Xn } is positive recurrent when μ 1; is null recurrent when μ = 1; and is transient when μ 1. In this paper, the integrability of the first returning time and the last exit time are discussed.  相似文献   

3.
In this paper we extend and improve some results of the large deviation for random sums of random variables. Let {Xn;n 〉 1} be a sequence of non-negative, independent and identically distributed random variables with common heavy-tailed distribution function F and finite mean μ ∈R^+, {N(n); n ≥0} be a sequence of negative binomial distributed random variables with a parameter p C (0, 1), n ≥ 0, let {M(n); n ≥ 0} be a Poisson process with intensity λ 〉 0. Suppose {N(n); n ≥ 0}, {Xn; n≥1} and {M(n); n ≥ 0} are mutually independent. Write S(n) =N(n)∑i=1 Xi-cM(n).Under the assumption F ∈ C, we prove some large deviation results. These results can be applied to certain problems in insurance and finance.  相似文献   

4.
Let C be a closed convex weakly Cauchy subset of a normed space X. Then we define a new {a,b,c} type nonexpansive and {a,b,c} type contraction mapping T from C into C. These types of mappings will be denoted respectively by {a,b,c}-ntype and {a,b,c}-ctype. We proved the following: 1. If T is {a,b,c}-ntype mapping, then inf{ || T(x)-x|| :x C C} =0, accordingly T has a unique fixed point. Moreover, any sequence {Xn}n∈NN in C with limn→∞||T(xn) - Xn|| = 0 has a subsequence strongly convergent to the unique fixed point of T. 2. If T is {a,b,c}-ctype mapping, then T has a unique fixed point. Moreover, for any x∈C the sequence of iterates {Tn (x)}n∈N has subsequence strongly convergent to the unique fixed point of T. This paper extends and generalizes some of the results given in [2,4, 7] and [13].  相似文献   

5.
§ 1  Introduction and main resultsL et { X,Xn;n≥ 1} be a sequence of random variables with common distributionfunction F,mean0 and positive,finite variance,and set Sn= nk=1 Xk,n≥ 1.Also letlogx= ln(x∨e) ,log logx=log(logx) and(x) =2 xlog logx.Gut and Sp taru[2 ] studied theprecise asymptotics on the law of the iterated logarithm.One of their results is as follows.Theorem A.Spuuose that{ X ,Xn;n≥ 1} is a sequence of i.i.d.random variables with EX= 0 and0 相似文献   

6.
Let{Xn;n≥1}be a sequence of i.i.d, random variables with finite variance,Q(n)be the related R/S statistics. It is proved that lim ε↓0 ε^2 ∑n=1 ^8 n log n/1 P{Q(n)≥ε√2n log log n}=2/1 EY^2,where Y=sup0≤t≤1B(t)-inf0≤t≤sB(t),and B(t) is a Brownian bridge.  相似文献   

7.
Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1+X2+…+Xn,Mn=max k≤n│Sk│,n≥1.Let an=O(1/loglogn).In this paper,we prove that,for b〉-1,lim ε→0 →^2(b+1)∑n=1^∞ (loglogn)^b/nlogn n^1/2 E{Mn-σ(ε+an)√2nloglogn}+σ2^-b/(b+1)(2b+3)E│N│^2b+3∑k=0^∞ (-1)k/(2k+1)^2b+3 holds if and only if EX=0 and EX^2=σ^2〈∞.  相似文献   

8.
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.  相似文献   

9.
This paper is a further investigation of large deviation for partial and random sums of random variables, where {Xn,n ≥ 1} is non-negative independent identically distributed random variables with a common heavy-tailed distribution function F on the real line R and finite mean μ∈ R. {N(n),n ≥ 0} is a binomial process with a parameter p ∈ (0,1) and independent of {Xn,n ≥ 1}; {M(n),n ≥ 0} is a Poisson process with intensity λ 〉 0, Sn = ΣNn i=1 Xi-cM(n). Suppose F ∈ C, we futher extend and improve some large deviation results. These results can apply to certain problems in insurance and finance.  相似文献   

10.
We consider a Markov chain X = {Xi,i = 1,2,...} with the state space {0,1},and define W =∑i=1n XiXi+1,which is the number of 2-runs in X before time n + 1.In this paper,we prove that the negative binomial distribution is an appropriate approximation for LW when VarW is greater than EW.The error estimate obtained herein improves the corresponding result in previous literatures.  相似文献   

11.
12.
We consider a sequence X 1, ..., X n of r.v.'s generated by a stationary Markov chain with state space A = {0, 1, ..., r}, r 1. We study the overlapping appearances of runs of k i consecutive i's, for all i = 1, ..., r, in the sequence X 1,..., X n. We prove that the number of overlapping appearances of the above multiple runs can be approximated by a Compound Poisson r.v. with compounding distribution a mixture of geometric distributions. As an application of the previous result, we introduce a specific Multiple-failure mode reliability system with Markov dependent components, and provide lower and upper bounds for the reliability of the system.  相似文献   

13.
Let {(X i,Z i)} be an i.i.d. sequence of random pairs in a finite set × x ℒ; we will call it a discrete memoryless stationary correlated (DMSC) source with generic distribution dist(X 1,Z 1). Two DMSC sources {(X i,Z i)} and {(X i′,Z i′)} are called asymptotically isomorphic in the weak sense if for every ε>0 and sufficiently largen, there exists a joint distribution dist(X n,Z n,X′ n,Z′ n) ofn-length blocks of the two sources such that . For single sources of equal entropy, McMillan’s theorem implies asymptotic isomorphy in the sense suggested by this definition. For correlated sources, however, no nontrivial cases of weak asymptotic isomorphy are known. We show that some spectral properties of the generic distributions are invariant for weak asymptotic isomorphy, and these properties wholly determine the generic distribution in many cases.  相似文献   

14.
Summary Let {X n,j,−∞<j<∞∼,n≧1, be a sequence of stationary sequences on some probability space, with nonnegative random variables. Under appropriate mixing conditions, it is shown thatS n=Xn,1+…+X n,n has a limiting distribution of a general infinitely divisible form. The result is applied to sequences of functions {f n(x)∼ defined on a stationary sequence {X j∼, whereX n.f=fn(Xj). The results are illustrated by applications to Gaussian processes, Markov processes and some autoregressive processes of a general type. This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship of the National Sciences Foundation, Grant MCS 82-01119.  相似文献   

15.
Summary Let the random variablesX 1,X 2, ...,X n be generated by the first-order autoregressive modelX i =θX i−1 +e i wheree i ,i=1, 2, ...,n, are i.i.d. random variables with mean zero, variance σ2, and with unspecified density functiong(·). In the present paper we obtain a characterization of limiting distributions of nonparametric and parametric estimators of θ as well as a local asymptotic minimax bound of the risks of estimators.  相似文献   

16.
Fernando Szechtman 《代数通讯》2013,41(11):4973-4985
Let f(Z) = Zn ? a1Zn?1 + … + (?1)n?1an?1Z + (?1)nan be a monic polynomial with coefficients in a ring R with identity, not necessarily commutative. We study the ideal If of R[X1,…, Xn] generated by σi(X1,…, Xn) ? ai, where σ1,…, σn are the elementary symmetric polynomials, as well as the quotient ring R[X1,…, Xn]/If.  相似文献   

17.
A system s{ X(t)} = {X 1(t),X 2(t),..., X N(t)} of N interacting time reversible continuous time Markov chains is considered. The state space of each of the processes {X i(t)} (i = 1, 2,...,N) is partitioned into two aggregates. Interaction between the processes {X i(t)},{X 2(t)},...,{X N(t)} is introduced by allowing the transition rates of an individual process at time t to depend on the configuration of aggregates occupied by the other N - 1 processes at that time. The motivation for this work comes from ion channel modeling, where {(X}(t)} describes the gating mechanisms of N channels and the partitioning of the state space of {X i(t)} correspond to whether the channel is conducting or not. Let S(t) denote the number of conducting channels at time t. For a time-reversible class of such processes, expressions are derived for the mean and probability density function of the sojourns of {S(t)} at its different levels when {X(t)} is in equilibrium. Particular attention is paid to the situation when the N channels are located on a circle with nearest neighbor interaction. Necessary and sufficient conditions for a general co-operative multiple channel system to be time reversible are derived.  相似文献   

18.
We consider asymptotic expansions for sums Sn on the form Sn = ƒ0(X0) + ƒ(X1, X0) + … + ƒ(Xn, Xn−1), where Xi is a Markov chain. Under different ergodicity conditions on the Markov chain and certain conditional moment conditions on ƒ(Xi, Xi−1), a simple representation of the characteristic function of Sn is obtained. The representation is in term of the maximal eigenvalue of the linear operator sending a function g(x) into the function xE(g(Xi)exp[itƒ(Xi, x)]|Xi−1 = x).  相似文献   

19.
Let X,X 1,X 2, … be independent identically distributed random variables, F(x) = P{X < x}, S 0 = 0, and S n i=1 n X i . We consider the random variables, ladder heights Z + and Z that are respectively the first positive sum and the first negative sum in the random walk {S n }, n = 0, 1, 2, …. We calculate the first three (four in the case EX = 0) moments of random variables Z + and Z in the qualitatively different cases EX > 0, EX < 0, and EX = 0. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 159–179, April–June, 2006.  相似文献   

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