• Aucun résultat trouvé

Erratum to: The hitting distributions of a half real line for two-dimensional random

N/A
N/A
Info
Télécharger
Protected

Academic year: 2022

Partager "Erratum to: The hitting distributions of a half real line for two-dimensional random"

Copied!
2
0
0

Chargement.... (Voir le texte intégral maintenant)

Texte intégral

(1)

Ark. Mat.,50(2012), 199–200 DOI: 10.1007/s11512-011-0162-4

c 2012 by Institut Mittag-Leffler. All rights reserved

Erratum to: The hitting distributions of a half real line for two-dimensional random

walks

Kˆohei Uchiyama

The original article appeared inArk. Mat.48(2010), 371–393, doi:10.1007/s11512-009-0096-2.

In Theorem 1.1 of my paper [1] (p. 373) there is an erroneous statement. The formula (3) of the theorem expresses the asymptotic form of Hx−(s), the hitting distribution of the non-positive half line for a random walk onZ2 started atx. Its first statement, which is true, asserts that (3) holds forx>0. The error is included in the second one, in which the validity of (3) is asserted also forx<0 under the additional moment conditionE[|S(1)|2log|S(1)|]<∞. Forx<0 however, the right- hand side of (3) must be multiplied by the ratio (|x|+|s|)/2|x−s|, namely the true statement must read: ifE[|S(1)|2log|S(1)|]<∞in addition, then forx<0,

(3) Hx−(s) =σ2 π

(|x|+|s|)ν(x)μ(s)

2|x−s|2 (1+o(1)).

The statement of Theorem 1.1 for the case x<0 without the additional moment condition remains true, since it imposes a restriction on the manner of|x−s|tending to∞that entails eitherx/s→0 ors/x→0, when (3) and (3) agree.

The formula (3) is obtained by setting y=0 in the formula of Theorem 1.4 (p. 376). Although this special case is excluded in Theorem 1.4, it is verified essen- tially along the lines preceding it (see Section 5 of [2] for some ingredients of the proof that are not mentioned in [1]).

In addition there are simple errors on p. 376: the factor (1+o(1)) is missing from the right-hand sides of (14) and the formula of Theorem 1.4.

The online version of the original article can be found under doi:10.1007/s11512-009-0096-2.

(2)

200 Kˆohei Uchiyama:

Erratum to: The hitting distributions of a half real line for two-dimensional random walks

References

1. Uchiyama, K., The hitting distributions of a half real line for two-dimensional random walks,Ark.Mat.48(2010), 371–393.

2. Uchiyama, K., The hitting distributions of line segments for two dimensional random walks,Preprint, 2011.arXiv:1105.5863v1.

Kˆohei Uchiyama

Department of Mathematics Tokyo Institute of Technology Oh-okayama, Meguro

Tokyo 152-8551 Japan

uchiyama@math.titech.ac.jp

Received May 16, 2011

published online January 26, 2012

Références

Télécharger maintenant ( PDF - 2 Page - 277.56 KB )

Documents relatifs

(2)The third is a consequence of the latter, because it recognizes that existing surface water rules are not applicable to aquifers

2. Duty to harmonize the special or regional conventions with the basic principles of this Convention. It is a convenient flexibility to allow the amendment of the application of

In the paper [1], the proof of Theorem 7.10 is incomplete

In the cases which were considered in the proof of [1, Proposition 7.6], the case when the monodromy group of a rank 2 stable bundle is the normalizer of a one-dimensional torus

In this paper, it is proved that the conjecture is true if every nonzero complement left ideal of the ringR is not small (or not singular)

Suppose R is a right noetherian, left P-injective, and left min-CS ring such that every nonzero complement left ideal is not small (or not singular).. Then R

The hitting distributions of a half real line for two-dimensional random walks

For every two-dimensional random walk on the square lattice Z 2 having zero mean and finite variance we obtain fine asymptotic estimates of the probability that the walk hits the

In the number theory case, it is SpecZ that plays the role of the base curve

the one developed in [2, 3, 1] uses R -filtrations to interpret the arithmetic volume function as the integral of certain level function on the geometric Okounkov body of the

With all this interest, it is unsurprising that numerous proofs of the theorem have been discovered

The winding number of a closed curve around a given point is an integer representing the total number of times that curve travels anti-clockwise around the point.. The sign of

Now it is easy to see that ψ=χW+χW0, so it is equal to the character ofW⊕W0

(b) The previous question shows that it is enough to consider the case k, l 6 n/2, for otherwise we can replace the representations by isomorphic ones so that these inequalities

The writer learnt that, in both, it is not enough to have a health care system that treats the symptoms. It has to tackle the root cause of the problem.

Our research showed that an estimated quarter of a million people in Glasgow (84 000 homes) and a very conservative estimate of 10 million people nation- wide, shared our