The limit distribution of the largest interpoint distance for distributions supported by a d-dimensional ellipsoid and generalizations

2016 ◽  
Vol 48 (4) ◽  
pp. 1256-1270 ◽  
Author(s):  
Michael Schrempp
Keyword(s):  
Asymptotic Behaviour ◽  
Limit Distribution ◽  
General Setting ◽  
Major Axis ◽  
Fixed Number ◽  
Poisson Processes ◽  
Interpoint Distance

AbstractWe study the asymptotic behaviour of the maximum interpoint distance of random points in a d-dimensional ellipsoid with a unique major axis. Instead of investigating only a fixed number of n points as n tends to ∞, we consider the much more general setting in which the random points are the supports of appropriately defined Poisson processes. Our main result covers the case of uniformly distributed points.

On interpoint distances for planar Poisson cluster processes

1983 ◽  
Vol 20 (3) ◽  
pp. 513-528 ◽  
Author(s):  
Richard J. Kryscio ◽  
Roy Saunders
Keyword(s):  
Lebesgue Measure ◽  
Distance Function ◽  
Nearest Neighbor ◽  
Indicator Function ◽  
Poisson Processes ◽  
Interpoint Distance ◽  
Poisson Cluster ◽  
Cluster Processes

For stationary Poisson or Poisson cluster processes ξ on R2 we study the distribution of the interpoint distances using the interpoint distance function and the nearest-neighbor indicator function . Here Sr (x) is the interior of a circle of radius r having center x, I(t) is that subset of D which has x ∊ D and St(x) ⊂ D and χ is the usual indicator function. We show that if the region D ⊂ R2 is large, then these functions are approximately distributed as Poisson processes indexed by and , where µ(D) is the Lebesgue measure of D.

Components of Random Forests

1992 ◽  
Vol 1 (1) ◽  
pp. 35-52 ◽  
Author(s):  
Tomasz Łuczak ◽  
Boris Pittel
Keyword(s):  
Phase Transition ◽  
Random Forest ◽  
Asymptotic Behaviour ◽  
Random Graph ◽  
Random Forests ◽  
Limit Distribution ◽  
The Family ◽  
Supercritical Phase

A forest ℱ(n, M) chosen uniformly from the family of all labelled unrooted forests with n vertices and M edges is studied. We show that, like the Érdős-Rényi random graph G(n, M), the random forest exhibits three modes of asymptotic behaviour: subcritical, nearcritical and supercritical, with the phase transition at the point M = n/2. For each of the phases, we determine the limit distribution of the size of the k-th largest component of ℱ(n, M). The similarity to the random graph is far from being complete. For instance, in the supercritical phase, the giant tree in ℱ(n, M) grows roughly two times slower than the largest component of G(n, M) and the second largest tree in ℱ(n, M) is of the order n⅔ for every M = n/2 +s, provided that s3n−2 → ∞ and s = o(n), while its counterpart in G(n, M) is of the order n2s−2 log(s3n−2) ≪ n⅔.

On interpoint distances for planar Poisson cluster processes

1983 ◽  
Vol 20 (03) ◽  
pp. 513-528
Author(s):  
Richard J. Kryscio ◽  
Roy Saunders
Keyword(s):  
Lebesgue Measure ◽  
Distance Function ◽  
Nearest Neighbor ◽  
Indicator Function ◽  
Poisson Processes ◽  
Interpoint Distance ◽  
Poisson Cluster ◽  
Cluster Processes

For stationary Poisson or Poisson cluster processes ξ on R2 we study the distribution of the interpoint distances using the interpoint distance function and the nearest-neighbor indicator function . Here Sr (x) is the interior of a circle of radius r having center x, I(t) is that subset of D which has x ∊ D and St (x) ⊂ D and χ is the usual indicator function. We show that if the region D ⊂ R2 is large, then these functions are approximately distributed as Poisson processes indexed by and , where µ(D) is the Lebesgue measure of D.

Poisson approximation of intermediate empirical processes

1992 ◽  
Vol 29 (4) ◽  
pp. 825-837 ◽  
Author(s):  
E. Kaufmann ◽  
R.-D. Reiss
Keyword(s):  
Asymptotic Behaviour ◽  
Poisson Process ◽  
Sample Size ◽  
Poisson Approximation ◽  
Empirical Processes ◽  
Extreme Value ◽  
Poisson Processes ◽  
Homogeneous Poisson Process ◽  
Von Mises

We investigate the asymptotic behaviour of empirical processes truncated outside an interval about the (1 – s(n)/n)-quantile where s(n) → ∞ and s(n)/n → 0 as the sample size n tends to ∞. It is shown that extreme value (Poisson) processes and, alternatively, the homogeneous Poisson process may serve as approximations if certain von Mises conditions hold.

scholarly journals Approval-Based Apportionment

2020 ◽  
Vol 34 (02) ◽  
pp. 1854-1861
Author(s):  
Markus Brill ◽  
Paul Gölz ◽  
Dominik Peters ◽  
Ulrike Schmidt-Kraepelin ◽  
Kai Wilker
Keyword(s):  
Polynomial Time ◽  
General Setting ◽  
Fixed Number ◽  
Natural Restriction ◽  
Apportionment Problem

In the apportionment problem, a fixed number of seats must be distributed among parties in proportion to the number of voters supporting each party. We study a generalization of this setting, in which voters cast approval ballots over parties, such that each voter can support multiple parties. This approval-based apportionment setting generalizes traditional apportionment and is a natural restriction of approval-based multiwinner elections, where approval ballots range over individual candidates. Using techniques from both apportionment and multiwinner elections, we are able to provide representation guarantees that are currently out of reach in the general setting of multiwinner elections: First, we show that core-stable committees are guaranteed to exist and can be found in polynomial time. Second, we demonstrate that extended justified representation is compatible with committee monotonicity.

Compound Poisson approximation for long increasing sequences

2001 ◽  
Vol 38 (2) ◽  
pp. 449-463 ◽  
Author(s):  
Ourania Chryssaphinou ◽  
Eutichia Vaggelatou
Keyword(s):  
Asymptotic Behaviour ◽  
Poisson Distribution ◽  
Limit Distribution ◽  
Continuous Distribution ◽  
Poisson Approximation ◽  
Independent Random Variables ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Increasing Trend ◽  
Increasing Sequences

Consider a sequence X1,…,Xn of independent random variables with the same continuous distribution and the event Xi-r+1 < ⋯ < Xi of the appearance of an increasing sequence with length r, for i=r,…,n. Denote by W the number of overlapping appearances of the above event in the sequence of n trials. In this work, we derive bounds for the total variation and Kolmogorov distances between the distribution of W and a suitable compound Poisson distribution. Via these bounds, an associated theorem concerning the limit distribution of W is obtained. Moreover, using the previous results we study the asymptotic behaviour of the length of the longest increasing sequence. Finally, we suggest a non-parametric test based on W for checking randomness against local increasing trend.

Poisson approximation of intermediate empirical processes

1992 ◽  
Vol 29 (04) ◽  
pp. 825-837
Author(s):  
E. Kaufmann ◽  
R.-D. Reiss
Keyword(s):  
Asymptotic Behaviour ◽  
Poisson Process ◽  
Sample Size ◽  
Poisson Approximation ◽  
Empirical Processes ◽  
Extreme Value ◽  
Poisson Processes ◽  
Homogeneous Poisson Process ◽  
Von Mises

We investigate the asymptotic behaviour of empirical processes truncated outside an interval about the (1 – s(n)/n)-quantile where s(n) → ∞ and s(n)/n → 0 as the sample size n tends to ∞. It is shown that extreme value (Poisson) processes and, alternatively, the homogeneous Poisson process may serve as approximations if certain von Mises conditions hold.

Convergence in distribution of lightly trimmed and untrimmed sums are equivalent

1993 ◽  
Vol 113 (3) ◽  
pp. 615-638 ◽  
Author(s):  
Harry Kesten
Keyword(s):  
Limit Distribution ◽  
Random Variables ◽  
Fixed Number ◽  
Convergence In Distribution ◽  
Trimmed Sums ◽  
Limiting Behaviour ◽  
Modest Effect

AbstractWe show that trimming a fixed number of terms from sums of i.i.d. random variables (so-called light trimming) can have only a modest effect on limiting behaviour. More specifically, the trimmed sums, after centralization and normalization, have a limit distribution, if and only if the untrimmed sums have a limit distribution (with the same centralization and normalization constants).

Compound Poisson approximation for long increasing sequences

2001 ◽  
Vol 38 (02) ◽  
pp. 449-463
Author(s):  
Ourania Chryssaphinou ◽  
Eutichia Vaggelatou
Keyword(s):  
Asymptotic Behaviour ◽  
Poisson Distribution ◽  
Limit Distribution ◽  
Continuous Distribution ◽  
Poisson Approximation ◽  
Independent Random Variables ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Increasing Trend ◽  
Increasing Sequences

Consider a sequence X 1,…,X n of independent random variables with the same continuous distribution and the event X i-r+1 &lt; ⋯ &lt; X i of the appearance of an increasing sequence with length r, for i=r,…,n. Denote by W the number of overlapping appearances of the above event in the sequence of n trials. In this work, we derive bounds for the total variation and Kolmogorov distances between the distribution of W and a suitable compound Poisson distribution. Via these bounds, an associated theorem concerning the limit distribution of W is obtained. Moreover, using the previous results we study the asymptotic behaviour of the length of the longest increasing sequence. Finally, we suggest a non-parametric test based on W for checking randomness against local increasing trend.

Sign in / Sign up

Export Citation Format

Share Document