A law of the iterated logarithm for heavy-tailed random vectors

Scheffler, Hans–Peter

doi:10.1007/pl00008729

Cited by 12 publications

(11 citation statements)

References 20 publications

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“…It follows that (35) is indeed an upper bound of lim sup p→∞ P p (τ x = T m and K m > m). It is clear that the ratio in (35) converges to zero when m → ∞. Therefore This completes the proof.…”

Section: Convergence Towards P ∞mentioning

confidence: 89%

“…, then we would have T m = ∞ almost surely, and τ x < ∞ with high probability in the limit x → ∞ thanks to the law of iterated logarithm for heavy-tailed random walks [35]. The following lemma affirms that we can still use the function xf (n) to bound the deviation of L p (X n , Y n ) n≥0 up to time T m in the limit p → ∞ and when both x and m are large.…”

Section: The Markov Chain L P (P N ) N≥0mentioning

confidence: 99%

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Critical Ising Model on Random Triangulations of the Disk: Enumeration and Local Limits

Chen

Turunen

2020

Commun. Math. Phys.

134

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In [ ], we have studied the Boltzmann random triangulation of the disk coupled to an Ising model with Dobrushin boundary condition at its critical temperature. In this paper, we investigate the phase transition of this model by extending our previous results to arbitrary temperature: We compute the partition function of the model at all temperatures, and derive several critical exponents associated with the in nite perimeter limit. We show that the model has a local limit at any temperature, whose properties depend drastically on the temperature. At high temperatures, the local limit is reminiscent of the uniform in nite half-planar triangulation (UIHPT) decorated with a subcritical percolation. At low temperatures, the local limit develops a bottleneck of nite width due to the energy cost of the main Ising interface between the two spin clusters imposed by the Dobrushin boundary condition. This change can be summarized by a novel order parameter with a nice geometric meaning. In addition to the phase transition, we also generalize our construction of the local limit from the two-step asymptotic regime used in [ ] to a more natural diagonal asymptotic regime. We obtain in this regime a scaling limit related to the length of the main Ising interface, which coincides with predictions from the continuum theory of quantum surfaces (a.k.a. Liouville quantum gravity). their roots in the physics literature [ ], and was revisited and popularized by Angel in [ ]. The peeling process proves to be a valuable tool for understanding the geometry of random planar maps without Ising model, see [ ] for a review of recent developments.In a previous paper [ ], we extended some enumeration results of Bernardi and Bousquet-Mélou [ ] to study the Ising-decorated random triangulations with Dobrushin boundary condition at its critical temperature. We used the peeling process to construct the local limit of the model, and to obtain several scaling limit results concerning the lengths of some Ising interfaces. In this paper, we extend similar results to the model at any temperature, and show how the large scale geometry of Ising-decorated random triangulations changes qualitatively at the critical temperature. In particular, our results con rm the physical intuition that, at large scale, Ising-decorated random maps at non-critical temperatures behave like non-decorated random maps.A similar model of Ising-decorated triangulations has been studied in a recent work of Albenque, Ménard and Schae er [ ]. They followed an approach reminiscent of Angel and Schramm in [ ] to show that the model has a local limit at any temperature, and obtained several properties of the limit such as one-endedness and recurrence for some temperatures. However they studied the model without boundary, and hence did not encounter the geometric consequences of the phase transition.We start by recalling some essential de nitions from [ ].Planar maps. Recall that a ( nite) planar map is a proper embedding of a nite connected graph into the sphere S 2 , viewed up to o...

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Section: Convergence Towards P ∞mentioning

confidence: 89%

Section: The Markov Chain L P (P N ) N≥0mentioning

confidence: 99%

Critical Ising Model on Random Triangulations of the Disk: Enumeration and Local Limits

Chen

Turunen

2020

Commun. Math. Phys.

134

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“…Scheffler [34] showed that if X ∈ GDOSA(Y,c), then for all unit vectors θ ∈ R d (1.5) where {x n } is a sequence of positive numbers such that β −1 n x n → ∞. Here and in the sequel, f n g n means 0 < lim inf f n /g n ≤ lim sup f n /g n < ∞.…”

Section: Introductionmentioning

confidence: 97%

Large Deviations for Sums of Random Vectors Attracted to Operator Semi-Stable Laws

Wang

2015

J Theor Probab

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Let {X i , i ≥ 1} be i.i.d. R d -valued random vectors attracted to operator semi-stable laws and write S n = n i=1 X i . This paper investigates precise large deviations for both the partial sums S n and the random sums S N (t) , where N (t) is a counting process independent of the sequence {X i , i ≥ 1}. In particular, we show for all unit vectors θ the asymptoticswhich holds uniformly for x-region [γ n , ∞), where ·, · is the standard inner product on R d and {γ n } is some monotone sequence of positive numbers. As applications, the precise large deviations for random sums of real-valued random variables with regularly varying tails and R d -valued random vectors with weakly negatively associated occurrences are proposed. The obtained results improve some related classical ones.

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“…Since then, several papers have been devoted to develop the classical Chover LIL. See, for example, Hedye 1969 showed that (1.1) holds when X is in the domain of normal attraction of a nonnormal stable law, Pakshirajan and Vasudeva 1977 discussed the limit points of the sequence , Kuelbs and Kurtz 1974 obtained the classical Chover LIL in a Hilbert space setting, Chen 2002 obtained the classical Chover LIL for the weighed sums, Vasudeva 1984 , Qi and Cheng 1996 , Peng and Qi 2003 established the Chover LIL when X is in the domain of attraction of a nonnormal stable law, Scheffler 2000 studied the classical Chover LIL when X is in the generalized domain of operator semistable attraction of some nonnormal law, Chen and Hu 2012 extended the results of Kuelbs and Kurtz 1974 to an arbitrary real separable Banach space, and so on. It should be pointed out that the previous papers only gave sufficient conditions for the classical Chover LIL.…”

Section: Introductionmentioning

confidence: 99%

A characterization of Chover-type law of iterated logarithm

Chen

2014

SpringerPlus

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Let 0 < α ≤ 2 and − ∞ <β <∞. Let {Xn;n ≥ 1} be a sequence of independent copies of a real-valued random variable X and set Sn = X1+⋯+Xn, n ≥ 1. We say X satisfies the (α,β)-Chover-type law of the iterated logarithm (and write X∈CTLIL(α,β)) if almost surely. This paper is devoted to a characterization of X ∈CTLIL(α,β). We obtain sets of necessary and sufficient conditions for X∈CTLIL(α,β) for the five cases: α = 2 and 0 < β <∞, α = 2 and β = 0, 1<α<2 and −∞<β<∞, α = 1 and −∞ <β <∞, and 0 < α <1 and −∞ <β <∞. As for the case where α = 2 and −∞ <β <0, it is shown that X∉CTLIL(2,β) for any real-valued random variable X. As a special case of our results, a simple and precise characterization of the classical Chover law of the iterated logarithm (i.e., X∈CTLIL(α,1/α)) is given; that is, X∈CTLIL(α,1/α) if and only if where whenever 1< α ≤ 2.Mathematics Subject Classification (2000)Primary: 60F15; Secondary: 60G50

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A law of the iterated logarithm for heavy-tailed random vectors

Cited by 12 publications

References 20 publications

Critical Ising Model on Random Triangulations of the Disk: Enumeration and Local Limits

Critical Ising Model on Random Triangulations of the Disk: Enumeration and Local Limits

Large Deviations for Sums of Random Vectors Attracted to Operator Semi-Stable Laws

A characterization of Chover-type law of iterated logarithm

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