23,861 research outputs found
Behavior of the Euler scheme with decreasing step in a degenerate situation
The aim of this paper is to study the behavior of the weighted empirical
measures of the decreasing step Euler scheme of a one-dimensional diffusion
process having multiple invariant measures. This situation can occur when the
drift and the diffusion coefficient are vanish simultaneously. As a first step,
we give a brief description of the Feller's classification of the
one-dimensional process. We recall the concept of attractive and repulsive
boundary point and introduce the concept of strongly repulsive point. That
allows us to establish a classification of the ergodic behavior of the
diffusion. We conclude this section by giving necessary and sufficient
conditions on the nature of boundary points in terms of Lyapunov functions. In
the second section we use this characterization to study the decreasing step
Euler scheme. We give also an numerical example in higher dimension
The conduct of the sample average when the first moment is infinite
Many books about probability and statistics only mention the weak and the strong law of large numbers for samples from distributions with finite expectation. However, these laws also hold for distributions with infinite expectation and then the sample average has to go to infinity with increasing sample size.\ud
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Being curious about the way in which this would happen, we simulated increasing samples (up to n= 40000) from three distributions with infinite expectation. The results were somewhat surprising at first sight, but understandable after some thought. Most statisticians, when asked, seem to expect a gradual increase of the average with the size of the sample. So did we. In general, however, this proves to be wrong and for different parent distributions different types of conduct appear from this experiment.\ud
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The samples from the "absolute Cauchy"-distribution are most interesting from a practical point of view: the average takes a high jump from time to time and decreases in between. In practice it might well happen, that the observations causing the jumps would be discarded as outlying observations
Optimal Cobordisms between Torus Knots
We construct cobordisms of small genus between torus knots and use them to
determine the cobordism distance between torus knots of small braid index. In
fact, the cobordisms we construct arise as the intersection of a smooth
algebraic curve in with the unit 4-ball from which a 4-ball of
smaller radius is removed. Connections to the realization problem of
-singularities on algebraic plane curves and the adjacency problem for
plane curve singularities are discussed. To obstruct the existence of
cobordisms, we use Ozsv\'ath, Stipsicz, and Szab\'o's -invariant,
which we provide explicitly for torus knots of braid index 3 and 4.Comment: 24 pages, 7 figures. Version 3: Minor corrections, implementation of
referee's recommendations. Comments welcom
A sharp signature bound for positive four-braids
We provide the optimal linear bound for the signature of positive four-braids
in terms of the three-genus of their closures. As a consequence, we improve
previously known linear bounds for the signature in terms of the first Betti
number for all positive braid links. We obtain our results by combining bounds
for positive three-braids with Gordon and Litherland's approach to signature
via unoriented surfaces and their Goeritz forms. Examples of families of
positive four-braids for which the bounds are sharp are provided.Comment: 12 pages, 6 figures, comments welcome! Accepted for publication in Q.
J. Mat
Markov chains conditioned never to wait too long at the origin
Motivated by Feller's coin-tossing problem, we consider the problem of conditioning an irreducible Markov chain never to wait too long at 0. Denoting by τ the first time that the chain, X, waits for at least one unit of time at the origin, we consider conditioning the chain on the event (τ›T). We show that there is a weak limit as T→∞ in the cases where either the state space is finite or X is transient. We give sufficient conditions for the existence of a weak limit in other cases and show that we have vague convergence to a defective limit if the time to hit zero has a lighter tail than τ and τ is subexponential
Balanced algebraic unknotting, linking forms, and surfaces in three- and four-space
We provide three 3-dimensional characterizations of the Z-slice genus of a
knot, the minimum genus of a locally-flat surface in 4-space cobounding the
knot whose complement has cyclic fundamental group: in terms of balanced
algebraic unknotting, in terms of Seifert surfaces, and in terms of
presentation matrices of the Blanchfield pairing. This result generalizes to a
knot in an integer homology 3-sphere and surfaces in certain simply connected
signature zero 4-manifolds cobounding this homology sphere. Using the
Blanchfield characterization, we obtain effective lower bounds for the Z-slice
genus from the linking pairing of the double branched cover of the knot. In
contrast, we show that for odd primes p, the linking pairing on the first
homology of the p-fold branched cover is determined up to isometry by the
action of the deck transformation group on said first homology.Comment: 39 pages, 5 figures, comments are welcome! v2: Added generalization
of the main theorem to knots and surfaces in more general 3- and 4-manifolds;
added new corollary showing equality of the Z-slice genus and the superslice
genus; expanded introduction, and added example in last sectio
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