On the distribution of values of sums of random variables

Chung, Kai Lai; Fuchs, Wolfgang

doi:10.1090/memo/0006

Cited by 162 publications

(140 citation statements)

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“…The discrete-time process {X n } ∞ n=1 is a one-dimensional mean-zero [in fact symmetric] random walk, which is necessarily recurrent thanks to the Chung-Fuchs theorem [8]. Consequently, the Lévy process {X t } t≥0 is recurrent as well.…”

Section: Proof Of Theorem 11: Lower Boundmentioning

confidence: 99%

On the existence and position of the farthest peaks of a family of stochastic heat and wave equations

Conus

Khoshnevisan

2010

Probab. Theory Relat. Fields

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We study the stochastic heat equation ∂ t u = Lu + σ(u)Ẇ in (1 + 1) dimensions, whereẆ is space-time white noise, σ : R → R is Lipschitz continuous, and L is the generator of a symmetric Lévy process that has finite exponential moments, and u 0 has exponential decay at ±∞. We prove that under natural conditions on σ: (i) The νth absolute moment of the solution to our stochastic heat equation grows exponentially with time; and (ii) The distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. Very little else seems to be known about the location of the high peaks of the solution to the stochastic heat equation under the present setting. [See, however,[19,20] for the analysis of the location of the peaks in a different model.]Finally, we show that these results extend to the stochastic wave equation driven by Laplacian.

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Section: Proof Of Theorem 11: Lower Boundmentioning

confidence: 99%

On the existence and position of the farthest peaks of a family of stochastic heat and wave equations

Conus

Khoshnevisan

2010

Probab. Theory Relat. Fields

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“…Let psx = the distribution of the first hit in (a+s, b + s) starting at x. For x large enough and |j| < / we have (1) J' \ps,x~Po,x\ < ia-(To see this let p be the distribution of the first hit in (a-s, b + s). Then |p-Po.…”

Section: Preliminariesmentioning

confidence: 99%

“…To see (1) note that for a given large number L we can choose v so that j/+b 2?=i T'r>L for all b by T0.4 (the integral does not depend on b). Next given e we can choose b so large that fl/2 J T Tr < - (2) For fixed C, j°_c 2¡"=i f\ is bounded independently of b.…”

Section: J ¡=Imentioning

confidence: 99%

Untitled

1969

Trans. Amer. Math. Soc.

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“…be independent, identically distributed random variables in Rk, k > 1, with common distribution F, and let for n > 1, S" = 2" X¡, S0 = 0. The random walk S = (SJ™ has a point of recurrence at x if, for every e > 0,It is well known that the set of recurrence points is either empty or equals the smallest closed additive group containing the support of F, see [1] or [3, §8.3]. In the latter case we say that S is recurrent.…”

mentioning

confidence: 99%

“…It is well known that the set of recurrence points is either empty or equals the smallest closed additive group containing the support of F, see [1] or [3, §8.3]. In the latter case we say that S is recurrent.…”

mentioning

confidence: 99%

On recurrence of a random walk in the plane

Chung¹,

Lindvall²

1980

Proc. Amer. Math. Soc.

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Abstract. The purpose of this note is to establish a sufficient condition for recurrence of a random walk (Sn) in R2. It follows from it that if S"/n1^2 is asymptotically normal then we have recurrence.Let Xx, X2,. . . be independent, identically distributed random variables in Rk, k > 1, with common distribution F, and let for n > 1, S" = 2" X¡, S0 = 0. The random walk S = (SJ™ has a point of recurrence at x if, for every e > 0,It is well known that the set of recurrence points is either empty or equals the smallest closed additive group containing the support of F, see [1] or [3, §8.3]. In the latter case we say that S is recurrent. Also well known is the following criterion: S is recurrent if and only if f P(\Sn\ 0 is sufficient for recurrence. In that paper it is also claimed that if, in R2, Sn/nx/2 is asymptotically normal, which is the case when E[X¡] = 0 (zero vector) and F^A^2] < oo, then we have recurrence. However, the argument indicated for this result in [2], and also in [3, Problem 14, p. 274], is misleading to say the least. This was discovered by students in Chung's class in 1977 and was first corrected by him then: it is the main purpose of this note to settle this matter.For y = (y" . . . ,yk) £ Rk, k > 2, we let \y\ = max,

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On the distribution of values of sums of random variables

Cited by 162 publications

References 0 publications

On the existence and position of the farthest peaks of a family of stochastic heat and wave equations

On the existence and position of the farthest peaks of a family of stochastic heat and wave equations

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On recurrence of a random walk in the plane

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