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Archive for Mathematical Logic - 相似文献
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Convergence of weighted averages of independent random variables 总被引:15,自引:0,他引:15
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A sequence (μ
n) of probability measures on the real line is said to converge vaguely to a measureμ if∫ fdμ
n →∫ fdμ for every continuous functionf withcompact support. In this paper one investigates problems analogous to the classical central limit problem under vague convergence.
Let ‖μ‖ denote the total mass ofμ andδ
0 denote the probability measure concentrated in the origin. For the theory of infinitesimal triangular arrays it is true in
the present context, as it is in the classical one, that all obtainable limit laws are limits of sequences of infinitely divisible
probability laws. However, unlike the classical situation, the class of infinitely divisible laws is not closed under vague
convergence. It is shown that for every probability measureμ there is a closed interval [0,λ], [0,e
−1] ⊂ [0,λ] ⊂ [0, 1], such thatβμ is attainable as a limit of infinitely divisible probability laws iffβ ε [0,λ]. In the independent identically distributed case, it is shown that if (x
1 + ... +x
n)/a
n, an → ∞, converges vaguely toμ with 0<‖μ‖<1, thenμ=‖μ‖δ
0. If furthermore the ratiosa
n+1/a
n are bounded above and below by positive numbers, thenL(x)=P[|X
1|>x] is a slowly varying function ofx. Conversely, ifL(x) is slowly varying, then for everyβ ε (0, 1) one can choosea
n → ∞ so that the limit measure=βδ
0.
To the memory of Shlomo Horowitz
This research was partially supported by the National Science Foundation. 相似文献
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For random walk on the d-dimensional integer lattice we consider again the problem of deciding when a set is recurrent, that is visited infinitely often with probability one by the random walk in question. Some special cases are considered, among them the following: for d = 2, what sequences (nj) have the property that with probability one the random walk visits the origin for infinitely many nj. A related problem, which is however not a special case of the recurrence problem, is to decide for what sequences (nj) the states visited by the random walk at times nj are all distinct, with only a finite number of exceptions. This problem is dealt with in the final part of the paper. 相似文献
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Let {S
n
, n=0, 1, 2, …} be a random walk (S
n
being thenth partial sum of a sequence of independent, identically distributed, random variables) with values inE
d
, thed-dimensional integer lattice. Letf
n
=Prob {S
1 ≠ 0, …,S
n
−1 ≠ 0,S
n
=0 |S
0=0}. The random walk is said to be transient if
and strongly transient if
. LetR
n
=cardinality of the set {S
0,S
1, …,S
n
}. It is shown that for a strongly transient random walk with p<1, the distribution of [R
n
−np]/σ √n converges to the normal distribution with mean 0 and variance 1 asn tends to infinity, where σ is an appropriate positive constant. The other main result concerns the “capacity” of {S
0, …,S
n
}. For a finite setA inE
d
, let C(A=Σ
x∈A
) Prob {S
n
∉A, n≧1 |S
0=x} be the capacity ofA. A strong law forC{S
0, …,S
n
} is proved for a transient random walk, and some related questions are also considered.
This research was partially supported by the National Science Foundation. 相似文献
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Steven Orey 《Probability Theory and Related Fields》1970,15(3):249-256
Summary Let X
t be a real Gaussian process with stationary increments, mean 0,
t
2
=E[(X
s+t–X
s)2] If
t
2
behaves like t
as t 0, 0<<1, the graph of a.e. sample function will have Hausdorff dimension 2 -. This leads one to feel that the set of zeros of X
t should have Hausdorff dimension 1 -. This is shown to be true provided the process is stationary and satisfies additional assumptions. 相似文献
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