Uniform asymptotic estimates of transition probabilities on combs

Bertacchi, Daniela; Zucca, Fabio

doi:10.1017/s1446788700008144

Cited by 31 publications

(34 citation statements)

References 16 publications

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“…Indeed in [3,Section 10], it has been remarked that, if k/n goes to zero with a certain speed, then p (2n) (2k, 0), (0, 0) /p (2n) (0, 2k), (0, 0) n→∞ −→ 0. Moreover the results in [3] imply that there are no sub-Gaussian estimate of the transition probabilities on C 2 . Such estimates have been found on many graphs: by Jones [12] on the 2-dimensional Sierpiński graph, by Barlow and Bass [1] on the graphical Sierpiński carpet and on rather general graphs by Grigor'yan and Telcs ([11, 18]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotic Behaviour of the Simple Random Walk on the 2-dimensional Comb

Bertacchi

2006

Electron. J. Probab.

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We analyze the differences between the horizontal and the vertical component of the simple random walk on the 2-dimensional comb. In particular we evaluate by combinatorial methods the asymptotic behaviour of the expected value of the distance from the origin, the maximal deviation and the maximal span in n steps, proving that for all these quantities the order is n 1/4 for the horizontal projection and n 1/2 for the vertical one (the exact constants are determined). Then we rescale the two projections of the random walk dividing by n 1/4 and n 1/2 the horizontal and vertical ones, respectively. The limit process is obtained. As a corollary of the estimate of the expected value of the maximal deviation, the walk dimension is determined, showing that the Einstein relation between the fractal, spectral and walk dimensions does not hold on the comb.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotic Behaviour of the Simple Random Walk on the 2-dimensional Comb

Bertacchi

2006

Electron. J. Probab.

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“…uniformly with respect to v ∈ V n (for the definition of uniform convergence with respect to a variable in a sequence of sets, see for instance [4 Since the interval…”

Section: The Upper Boundmentioning

confidence: 99%

Lower bounds for moments of global scores of pairwise Markov chains

Lember

Matzinger

Sova

et al. 2018

Stochastic Processes and their Applications

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be two random sequences so that every random variable takes values in a finite set A. We consider a global similarity score Ln := L(X 1 , . . . , Xn; Y 1 , . . . , Yn) that measures the homology (relatedness) of words (X 1 , . . . , Xn) and (Y 1 , . . . , Yn). A typical example of such score is the length of the longest common subsequence. We study the order of central absolute moment E|Ln − ELn| r in the case where the two-dimensional process (X 1 , Y 1 ), (X 2 , Y 2 ), . . . is a Markov chain on A × A. This is a very general model involving independent Markov chains, hidden Markov models, Markov switching models and many more. Our main result establishes a general condition that guarantees that E|Ln − ELn| r n r 2 . We also perform simulations indicating the validity of the condition.

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“…In what follows, we will disregard parity issues. Namely we will use results for the n−step transition probability for arbitrary n, u and v, when these results are only proved for even n, u and v. When these values are not even, and the corresponding probability is not zero, then their asymptotic behavior is the same as for even values (see Bertacchi and Zucca [4], Sections 3, 4 and 10).…”

Section: Remark 22mentioning

confidence: 99%