A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random Variables

Let X, X1, X2,… be a sequence of non-degenerate i.i.d. random variables with common distribution function F. For 1 ≤ j ≤ n, let mn(j) be the number of Xi satisfying either |Xi| > |Xj|, 1 ≤ i ≤ n, or |Xi| = |Xj|, 1 ≤ i ≤ j, and let (r)Xn = Xj if mn(j) = r. Thus (r)Xn is the rth largest random variable in absolute value from amongst X1, …, Xn with ties being broken according to the order in which the random variables occur. Set (r)Sn = (r+1)Xn + … + (n)Xn and write Sn for (0)Sn. We will refer to (r)Sn as a trimmed sum.

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Records from a power model of independent observations

Journal of Applied Probability ◽

10.2307/3215212 ◽

1995 ◽

Vol 32 (4) ◽

pp. 982-990 ◽

Cited By ~ 2

Author(s):

Ishay Weissman

Keyword(s):

Distribution Function ◽

Wide Spectrum ◽

Continuous Distribution ◽

Random Variables ◽

Continuous Distribution Function ◽

Power Model ◽

Record Times ◽

Independent Sequence ◽

Independent Observations

Records from are analyzed, where {Yj} is an independent sequence of random variables. Each Yj has a continuous distribution function Fj = Fλj for some distribution F and some λ j > 0. We study records, record times and related quantities for this sequence. Depending on the sequence of powers , a wide spectrum of behaviour is exhibited.

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On the law of the iterated logarithm for inter-record times

Journal of Applied Probability ◽

10.1017/s0021900200034999 ◽

1970 ◽

Vol 7 (02) ◽

pp. 432-439 ◽

Cited By ~ 10

Author(s):

William E. Strawderman ◽

Paul T. Holmes

Keyword(s):

Distribution Function ◽

Probability Space ◽

Continuous Distribution ◽

Random Variables ◽

Continuous Distribution Function ◽

Record Times ◽

Iterated Logarithm ◽

The Law ◽

Image Position ◽

Independent Identically Distributed

Let X 1, X2, X 3 , ··· be independent, identically distributed random variables on a probability space (Ω, F, P); and with a continuous distribution function. Let the sequence of indices {Vr } be defined as Also define The following theorem is due to Renyi [5].

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On the number and sum of near-record observations

Advances in Applied Probability ◽

10.1017/s000186780000046x ◽

2005 ◽

Vol 37 (03) ◽

pp. 765-780 ◽

Cited By ~ 5

Author(s):

N. Balakrishnan ◽

A.G. Pakes ◽

A. Stepanov

Keyword(s):

Distribution Function ◽

Asymptotic Properties ◽

Continuous Distribution ◽

Random Variables ◽

Continuous Distribution Function ◽

Record Time ◽

Record Value ◽

Independent And Identically Distributed

Let X 1,X 2,… be a sequence of independent and identically distributed random variables with some continuous distribution function F. Let L(n) and X(n) denote the nth record time and the nth record value, respectively. We refer to the variables X i as near-nth-record observations if X i ∈(X(n)-a,X(n)], with a>0, and L(n)<i<L(n+1). In this work we study asymptotic properties of the number of near-record observations. We also discuss sums of near-record observations.

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Linear processes are nearly Gaussian

Journal of Applied Probability ◽

10.2307/3212026 ◽

1967 ◽

Vol 4 (2) ◽

pp. 313-329 ◽

Cited By ~ 41

Author(s):

C. L. Mallows

Keyword(s):

Distribution Function ◽

Continuous Distribution ◽

Random Variables ◽

Linear Process ◽

Real Numbers ◽

Linear Processes ◽

Absolutely Continuous ◽

Restricted Sense ◽

Finite Set ◽

Third Moment

Let U denote the set of all integers, and suppose that Y = {Yu; u ∈ U} is a process of standardized, independent and identically distributed random variables with finite third moment and with a common absolutely continuous distribution function (d.f.) G (·). Let a = {au; u ∈ U} be a sequence of real numbers with Σuau2 = 1. Then Xu = ΣwawYu–w defines a stationary linear process X = {Xu; u ɛ U} with E(Xu) = 0, E(Xu2) = 1 for u ∊ U. Let F(·) be the d.f. of X0. We prove that if maxu |au| is small, then (i) for each w, Xw is close to Gaussian in the sense that ∫∞−∞(F(y) − Φ(y))2dy ≦ g maxu |au | where Φ(·) is the standard Gaussian d.f., and g depends only on G(·); (ii) for each finite set (w1, … wn), (Xw1, … Xwn) is close to Gaussian in a similar sense; (iii) the process X is close to Gaussian in a somewhat restricted sense. Several properties of the measures of distance from Gaussianity employed are investigated, and the relation of maxu|au| to the bandwidth of the filter a is studied.

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THE INVARIANCE PRINCIPLE FOR LINEAR PROCESSES WITH APPLICATIONS

Econometric Theory ◽

10.1017/s0266466602181072 ◽

2002 ◽

Vol 18 (1) ◽

pp. 119-139 ◽

Cited By ~ 17

Author(s):

Qiying Wang ◽

Yan-Xia Lin ◽

Chandra M. Gulati

Keyword(s):

Wiener Process ◽

Invariance Principle ◽

Random Variables ◽

Linear Process ◽

Standard Wiener Process ◽

Test Statistic ◽

Linear Processes ◽

Partial Sum ◽

Unit Root Testing ◽

Partial Sum Process

Let Xt be a linear process defined by Xt = [sum ]k=0∞ ψkεt−k, where {ψk, k ≥ 0} is a sequence of real numbers and {εk, k = 0,±1,±2,...} is a sequence of random variables. Two basic results, on the invariance principle of the partial sum process of the Xt converging to a standard Wiener process on [0,1], are presented in this paper. In the first result, we assume that the innovations εk are independent and identically distributed random variables but do not restrict [sum ]k=0∞ |ψk| < ∞. We note that, for the partial sum process of the Xt converging to a standard Wiener process, the condition [sum ]k=0∞ |ψk| < ∞ or stronger conditions are commonly used in previous research. The second result is for the situation where the innovations εk form a martingale difference sequence. For this result, the commonly used assumption of equal variance of the innovations εk is weakened. We apply these general results to unit root testing. It turns out that the limit distributions of the Dickey–Fuller test statistic and Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) test statistic still hold for the more general models under very weak conditions.

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Waitingtimes between record observations

Journal of Applied Probability ◽

10.1017/s0021900200025377 ◽

1967 ◽

Vol 4 (01) ◽

pp. 206-208 ◽

Cited By ~ 14

Author(s):

Marcel F. Neuts

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Asymptotic Normality ◽

Central Limit ◽

Law Of Large Numbers ◽

Continuous Distribution ◽

Random Variables ◽

Large Numbers ◽

Upper Record ◽

Independent Identically Distributed

If Δ r denotes the waitingtime between the (r − 1)st and the rth upper record in a sequence of independent, identically distributed random variables with a continuous distribution, then it is shown that Δ r satisfies the weak law of large numbers and a central limit theorem. This theorem supplements those of Foster and Stuart and Rényi, who investigated the index Vr of the rth upper record. Qualitatively the theorems establish the intuitive fact that for higher records, the waitingtime between the last two records outweighs even the total waitingtime for previous records. This explains also why the asymptotic normality of logVr is very inadequate for approximation purposes—Barton and Mallows.

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On record values and the exponent of a distribution with regularly varying upper tail

Journal of Applied Probability ◽

10.2307/3214894 ◽

1992 ◽

Vol 29 (3) ◽

pp. 575-586 ◽

Cited By ~ 11

Author(s):

Mohamed Berred

Keyword(s):

Distribution Function ◽

Asymptotic Behaviour ◽

Continuous Distribution ◽

Random Variables ◽

Record Values ◽

Numerical Results ◽

Continuous Distribution Function ◽

Regularly Varying

Let {Xn, n ≧ 1} be an i.i.d. sequence of positive random variables with a continuous distribution function having a regularly varying upper tail. Denote by {X(n), n ≧ 1} the corresponding sequence of record values. We introduce two statistics based on the sequence of successive record values and investigate their asymptotic behaviour. We also give some numerical results.

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On record values and the exponent of a distribution with regularly varying upper tail

Journal of Applied Probability ◽

10.1017/s0021900200043394 ◽

1992 ◽

Vol 29 (03) ◽

pp. 575-586 ◽

Cited By ~ 1

Author(s):

Mohamed Berred

Keyword(s):

Distribution Function ◽

Asymptotic Behaviour ◽

Continuous Distribution ◽

Random Variables ◽

Record Values ◽

Numerical Results ◽

Continuous Distribution Function ◽

Regularly Varying

Let {Xn, n ≧ 1} be an i.i.d. sequence of positive random variables with a continuous distribution function having a regularly varying upper tail. Denote by {X(n), n ≧ 1} the corresponding sequence of record values. We introduce two statistics based on the sequence of successive record values and investigate their asymptotic behaviour. We also give some numerical results.

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On the number and sum of near-record observations

Advances in Applied Probability ◽

10.1239/aap/1127483746 ◽

2005 ◽

Vol 37 (3) ◽

pp. 765-780 ◽

Cited By ~ 25

Author(s):

N. Balakrishnan ◽

A.G. Pakes ◽

A. Stepanov

Keyword(s):

Distribution Function ◽

Asymptotic Properties ◽

Continuous Distribution ◽

Random Variables ◽

Continuous Distribution Function ◽

Record Time ◽

Record Value ◽

Independent And Identically Distributed

Let X1,X2,… be a sequence of independent and identically distributed random variables with some continuous distribution function F. Let L(n) and X(n) denote the nth record time and the nth record value, respectively. We refer to the variables Xi as near-nth-record observations if Xi∈(X(n)-a,X(n)], with a>0, and L(n)<i<L(n+1). In this work we study asymptotic properties of the number of near-record observations. We also discuss sums of near-record observations.

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