On the existence of submultiplicative moments for the stationary distributions of some Markovian random walks

This paper is concerned with submultiplicative moments for the stationary distributions π of some Markov chains taking values in ℝ+ or ℝ which are closely related to the random walks generated by sequences of independent identically distributed random variables. Necessary and sufficient conditions are given for ∫φ(x)π(dx) < ∞, where φ(x) is a submultiplicative function, i.e. φ(0) = 1 and φ(x+y) ≤ φ(x)φ(y) for all x, y.

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On the existence of submultiplicative moments for the stationary distributions of some Markovian random walks

Journal of Applied Probability ◽

10.1239/jap/1032374230 ◽

1999 ◽

Vol 36 (1) ◽

pp. 78-85 ◽

Cited By ~ 1

Author(s):

M. S. Sgibnev

Keyword(s):

Markov Chains ◽

Random Walks ◽

Sufficient Conditions ◽

Random Variables ◽

Necessary And Sufficient Conditions ◽

Stationary Distributions ◽

Necessary And Sufficient ◽

Submultiplicative Function ◽

Independent Identically Distributed

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Moments for first-passage and last-exit times, the minimum, and related quantities for random walks with positive drift

Advances in Applied Probability ◽

10.1017/s0001867800016189 ◽

1986 ◽

Vol 18 (04) ◽

pp. 865-879 ◽

Cited By ~ 3

Author(s):

Svante Janson

Keyword(s):

Random Walks ◽

Sufficient Conditions ◽

Random Variables ◽

Necessary And Sufficient Conditions ◽

Partial Sums ◽

Exit Times ◽

First Passage ◽

Partial Sum ◽

Positive Expectation ◽

Necessary And Sufficient

Consider the sequence of partial sums of a sequence of i.i.d. random variables with positive expectation. We study various random quantities defined by the sequence of partial sums, e.g. the time at which the first or last crossing of a given level occurs, the value of the partial sum immediately before or after the crossing, the minimum of all partial sums. Necessary and sufficient conditions are given for the existence of moments of these quantities.

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Moments for first-passage and last-exit times, the minimum, and related quantities for random walks with positive drift

Advances in Applied Probability ◽

10.2307/1427253 ◽

1986 ◽

Vol 18 (4) ◽

pp. 865-879 ◽

Cited By ~ 24

Author(s):

Svante Janson

Keyword(s):

Random Walks ◽

Sufficient Conditions ◽

Random Variables ◽

Necessary And Sufficient Conditions ◽

Partial Sums ◽

Exit Times ◽

First Passage ◽

Partial Sum ◽

Positive Expectation ◽

Necessary And Sufficient

Consider the sequence of partial sums of a sequence of i.i.d. random variables with positive expectation.We study various random quantities defined by the sequence of partial sums, e.g. the time at which the first or last crossing of a given level occurs, the value of the partial sum immediately before or after the crossing, the minimum of all partial sums. Necessary and sufficient conditions are given for the existence of moments of these quantities.

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On the asymptotic normality of self-normalized sums

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100070018 ◽

1991 ◽

Vol 109 (3) ◽

pp. 597-610 ◽

Cited By ~ 24

Author(s):

Philip S. Griffin ◽

David M. Mason

Keyword(s):

Asymptotic Normality ◽

Sufficient Conditions ◽

Random Variables ◽

Necessary And Sufficient Conditions ◽

Necessary And Sufficient ◽

Independent Identically Distributed

AbstractLet X1, …, Xn be a sequence of non-degenerate, symmetric, independent identically distributed random variables, and let Sn(rn) denote their sum when the rn largest in modulus have been removed. We obtain necessary and sufficient conditions for asymptotic normality of the studentized version of Sn(rn), and compare this to the condition for asymptotic normality of the scalar normalized version. In particular, when rn = r these conditions are the same, but when rn → ∞the former holds more generally.

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On the law of the iterated logarithm in the infinite variance case

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700021868 ◽

1980 ◽

Vol 30 (1) ◽

pp. 5-14 ◽

Cited By ~ 2

Author(s):

R. A. Maller

Keyword(s):

Sufficient Conditions ◽

Random Variables ◽

Necessary And Sufficient Conditions ◽

Infinite Variance ◽

Iterated Logarithm ◽

The Law ◽

Necessary And Sufficient ◽

Independent And Identically Distributed

AbstractThe main purpose of the paper is to give necessary and sufficient conditions for the almost sure boundedness of (Sn – αn)/B(n), where Sn = X1 + X2 + … + XmXi being independent and identically distributed random variables, and αnand B(n) being centering and norming constants. The conditions take the form of the convergence or divergence of a series of a geometric subsequence of the sequence P(Sn − αn > a B(n)), where a is a constant. The theorem is distinguished from previous similar results by the comparative weakness of the subsidiary conditions and the simplicity of the calculations. As an application, a law of the iterated logarithm general enough to include a result of Feller is derived.

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NECESSARY AND SUFFICIENT CONDITIONS FOR GLOBAL GEOMETRIC CONVERGENCE OF BLOCK GAUSS-SEIDEL ITERATION ALGORITHM APPLIED TO MARKOV CHAINS

Journal of the Operations Research Society of Japan ◽

10.15807/jorsj.40.283 ◽

1997 ◽

Vol 40 (3) ◽

pp. 283-293

Author(s):

Ushio Sumita ◽

Nobuko Igaki

Keyword(s):

Markov Chains ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Iteration Algorithm ◽

Geometric Convergence ◽

Necessary And Sufficient

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Poisson theorem for non-homogeneous Markov chains

Journal of Applied Probability ◽

10.1017/s0021900200038237 ◽

1989 ◽

Vol 26 (03) ◽

pp. 637-642 ◽

Cited By ~ 1

Author(s):

Janusz Pawłowski

Keyword(s):

Markov Chains ◽

Poisson Distribution ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Compound Poisson Distribution ◽

Convergence In Distribution ◽

Compound Poisson ◽

Necessary And Sufficient

This paper gives necessary and sufficient conditions for the convergence in distribution of sums of the 0–1 Markov chains to a compound Poisson distribution.

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Stability of maxima of random variables defined on a Markov chain

Advances in Applied Probability ◽

10.2307/1426000 ◽

1972 ◽

Vol 4 (2) ◽

pp. 285-295 ◽

Cited By ~ 13

Author(s):

Sidney I. Resnick

Keyword(s):

Markov Chain ◽

Regularity Condition ◽

Sufficient Conditions ◽

Random Variables ◽

Necessary And Sufficient Conditions ◽

Finite Markov Chain ◽

Finite Product ◽

Normalizing Constants ◽

Necessary And Sufficient

Consider maxima Mn of a sequence of random variables defined on a finite Markov chain. Necessary and sufficient conditions for the existence of normalizing constants Bn such that are given. The problem can be reduced to studying maxima of i.i.d. random variables drawn from a finite product of distributions πi=1mHi(x). The effect of each factor Hi(x) on the behavior of maxima from πi=1mHi is analyzed. Under a mild regularity condition, Bn can be chosen to be the maximum of the m quantiles of order (1 - n-1) of the H's.

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