By Andrew Barbour, Hock Peng Chan, David Siegmund
In June 2010, a convention, Probability Approximations and Beyond, used to be held on the nationwide college of Singapore (NUS), in honor of pioneering mathematician Louis Chen. Chen made the 1st of a number of seminal contributions to the idea and alertness of Stein’s process. one in all his most vital contributions has been to show Stein’s focus inequality concept into a good software for supplying mistakes bounds for the traditional approximation in lots of settings, and specifically for sums of random variables displaying in simple terms neighborhood dependence. This convention attracted a wide viewers that got here to pay homage to Chen and to listen to displays by way of colleagues who've labored with him in targeted methods during the last forty+ years.
The papers during this quantity attest to how Louis Chen’s state of the art principles encouraged and proceed to steer such components as molecular biology and machine technological know-how. He has built purposes of his paintings on Poisson approximation to difficulties of sign detection in computational biology. the unique papers contained during this booklet supply historic context for Chen’s paintings along observation on a few of his significant contributions by way of noteworthy statisticians and mathematicians operating at the present time.
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Example text
11) The last “ ” holds because all the (Dn,i × Dn, j )’s are disjoint and contained in E ∩ D 2 and P(Wn = 0) P(N (D) = 0). En ∩ D 2 . It is easy to check that En ∩ D 2 Selecting a subsequence if necessary, we may assume that θn → θ. Then θ μ(D) because θn μ(D) for all n. 11). -L. Lan et al. 3 Proofs of the Main Results Let N = i δ X i be a homogeneous Poisson point process in R2 with intensity λ and C (λ, r ) = i B(X i , r ) be defined as in the beginning of Sect. 1 Observe that the randomly positioned discs divide the plane into random covered and uncovered regions.
5) with A = {r }; that frm (m) = 0 is given. 3) to obtain the second equality, we have αx frm (x + 2) − βx frm (x) = αx − π m ([0, x] ∩ L m,n )πrm m βx+2 πx+2 − βx − π m ([0, x − 2] ∩ L m,n )πrm βx πxm = αx − π m ([0, x] ∩ L m,n )πrm αx πxm − βx − π m ([0, x − 2] ∩ L m,n )πrm βx πxm = −πrm . 38 L. Goldstein and A. Xia If x = r then αx frm (x + 2) − βx frm (x) = αr π m ([r + 2, n] ∩ L m,n )πrm βr +2 πrm+2 − βr −π m ([0, r − 2] ∩ L m,n )πrm βr πrm = αr π m ([r + 2, n] ∩ L m,n )πrm αr πrm − βr −π m ([0, r − 2] ∩ L m,n )πrm βr πrm = π m ([r + 2, n] ∩ L m,n ) + π m ([0, r − 2] ∩ L m,n ) = 1 − πrm .
A general description of coverage process was introduced in [1] as follows: Let P ≡ {ξ1 , ξ2 , . } be a countable collection of points in k-dimensional Euclidean space, and {S1 , S2 , . } be a countable collection of non-empty sets. Define ξi +Si to be the set {ξi +x : x ∈ Si }. Then the set {ξi + Si : i = 1, 2, . } is called a coverage process. -L. -M. -Y. -Y. cn A. D. Barbour et al. -L. Lan et al. P may be a stochastic point process, and {Si } may be random sets. Our motivation of considering aforementioned special coverage process arises from the coverage problem of wireless sensor networks.