242 research outputs found
One-Dimensional Diffusions That Eventually Stop Down-Crossing
Consider a diffusion process corresponding to the operator
and which is transient to .
For , we give an explicit criterion in terms of the coefficients and
which determines whether or not the diffusion almost surely eventually
stops making down-crossings of length . As a particular case, we show that
if , then the diffusion almost surely stops making down-crossings of
length if , for some
and for large , but makes down-crossings of length at
arbitrarily large times if , for
large
When the law of large numbers fails for increasing subsequences of random permutations
Let the random variable denote the number of increasing
subsequences of length in a random permutation from , the symmetric
group of permutations of . In a recent paper [Random Structures
Algorithms 29 (2006) 277--295] we showed that the weak law of large numbers
holds for if ; that is,
The
method of proof employed there used the second moment method and demonstrated
that this method cannot work if the condition does not hold.
It follows from results concerning the longest increasing subsequence of a
random permutation that the law of large numbers cannot hold for if
, with . Presumably there is a critical exponent
such that the law of large numbers holds if , with , and
does not hold if , for some .
Several phase transitions concerning increasing subsequences occur at ,
and these would suggest that . However, in this paper, we show that
the law of large numbers fails for if
. Thus, the critical exponent,
if it exists, must satisfy .Comment: Published at http://dx.doi.org/10.1214/009117906000000728 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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