One dimensional lattice random walks with absorption at a point/on a half line

2019

J Theor Probab

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running head: Scaling limits of random walk bridges key words: random walk on the integer lattice; conditioned to avoid a set; third moment; functional limit theorem; killing on a finite set; tightness of pinned walk; tunneling.Abstract This paper concerns a scaling limit of a one-dimensional random walk S x n started from x on the integer lattice conditioned to avoid a non-empty finite set A, the random walk being assumed to be irreducible and have zero mean. Suppose the variance σ 2 of the increment law is finite. Given positive constants b, c and T we consider the scaledat another point ≈ −c √ N at t = T and avoid A in between and discuss the functional limit of it as N → ∞. We show that it converges in law to a continuous process if u] to vary regularly as u → −∞ with exponent −β, 2 ≤ β ≤ 3 and show that it converges to a process which has one downward jump that clears the origin if β < 3; in case β = 3 there arises the same limit process as in case E[|S 1 | 3 ; S 1 < 0] < ∞. In case σ 2 = ∞ we consider the special case when S 1 belongs to the domain of attraction of a stable law of index 1 < α < 2 having no negative jumps and obtain analogous results.

Section: Preliminary Resultssupporting

confidence: 52%

Scaling Limits of Random Walk Bridges Conditioned to Avoid a Finite Set

2019

J Theor Probab

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“…To this end we follow the arguments given in the second half of Section 3 with ρ replaced by −σ (If δ = 2, we need to bring in the logarithmic factor in the numerator in the O term as in (9.4).) In view of the expansion σ [14]) the two formulae (9.5) and (9.7) are consistent. The last O terms in them seem to represent the correct order (i.e.…”

Section: Proof Of Theorem 13mentioning

confidence: 63%

“…The constant C * is nonnegative and vanishes if and only if the walk is continuous in the vertical direction (namely p((x 1 , x 2 )) = 0 if |x 2 | ≥ 2) [14]. See Section 9 for more details.…”

Section: (|S| > ε|N| |S| → ∞)mentioning

confidence: 99%

See 1 more Smart Citation

The hitting distributions of a line for two dimensional random walks

Trans. Amer. Math. Soc.

2009

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“…We follow the proof in [27] given to the corresponding result for case σ 2 < ∞. We employ the representation…”

Section: Upper and Lower Bounds Of Q Nmentioning

confidence: 99%

Asymptotically stable random walks of index 1<α<2 killed on a finite set

Stochastic Processes and their Applications

2019

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running head: random walks killed on a finite set key words: one dimensional random walk; first passage time; killed at the origin; in a domain of attraction, transition probability, stable process AMS Subject classification (2010): Primary 60G50, Secondary 60J45. AbstractFor a random walk on the integer lattice Z that is attracted to a strictly stable process with index α ∈ (1, 2) we obtain the asymptotic form of the transition probability for the walk killed when it hits a finite set. The asymptotic forms obtained are valid uniformly in a natural range of the space and time variables. The situation is relatively simple when the limit stable process has jumps in both positive and negative directions; in the other case when the jumps are one sided rather interesting matters are involved and detailed analyses are necessitated.