28,883 research outputs found
On the Conditional Distribution of a Multivariate Normal given a Transformation - the Linear Case
We show that the orthogonal projection operator onto the range of the adjoint
of a linear operator can be represented as where is an invertible
linear operator. Using this representation we obtain a decomposition of a
Normal random vector as the sum of a linear transformation of that is
independent of and an affine transformation of . We then use this
decomposition to prove that the conditional distribution of a Normal random
vector given a linear transformation is again a multivariate
Normal distribution. This result is equivalent to the well-known result that
given a -dimensional component of a -dimensional Normal random vector,
where , the conditional distribution of the remaining
-dimensional component is a -dimensional
multivariate Normal distribution, and sets the stage for approximating the
conditional distribution of given , where is a
continuously differentiable vector field.Comment: 2/6/18: Updated the proof of Theorem 4 & added a corollary. arXiv
admin note: text overlap with arXiv:1612.0121
A First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle
We study the probability distribution of the total displacement
of an -step run and tumble particle on a line, in presence of a
constant nonzero drive . While the central limit theorem predicts a standard
Gaussian form for near its peak, we show that for large positive and
negative , the distribution exhibits anomalous large deviation forms. For
large positive , the associated rate function is nonanalytic at a critical
value of the scaled distance from the peak where its first derivative is
discontinuous. This signals a first-order dynamical phase transition from a
homogeneous `fluid' phase to a `condensed' phase that is dominated by a single
large run. A similar first-order transition occurs for negative large
fluctuations as well. Numerical simulations are in excellent agreement with our
analytical predictions.Comment: 35 pages, 5 figures. An algebraic error in Appendix B of the previous
version of the manuscript has been corrected. A new argument for the location
of the transition is reported in Appendix B.
Inonu-Wigner Contractions of Kac-Moody Algebras
We discuss In\"on\"u-Wigner contractions of affine Kac-Moody algebras. We
show that the Sugawara construction for the contracted affine algebra exists
only for a fixed value of the level , which is determined in terms of the
dimension of the uncontracted part of the starting Lie algebra, and the
quadratic Casimir in the adjoint representation. Further, we discuss
contractions of coset spaces, and obtain an affine {\it translation}
algebra, which yields a Virasoro algebra (via a GKO construction) with a
central charge given by .Comment: 11 pages, IMSc/92-2
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