K R Parthasarathy
Articles written in Proceedings – Mathematical Sciences
Volume 104 Issue 1 February 1994 pp 253-262 Obituary note
Kolmogorov’s existence theorem for Markov processes in
B V Rajarama Bhat K R Parthasarathy
Given a family of transition probability functions between measure spaces and an initial distribution Kolmogorov’s existence theorem associates a unique Markov process on the product space. Here a canonical non-commutative analogue of this result is established for families of completely positive maps between
Volume 113 Issue 1 February 2003 pp 3-13
A remark on the unitary group of a tensor product of
Let
Volume 114 Issue 4 November 2004 pp 365-374 Regular Articles
On the maximal dimension of a completely entangled subspace for finite level quantum systems
Let
When$${\mathcal{H}} = {\mathcal{H}}_2 $$ and
We also introduce a more delicate notion of a
Volume 117 Issue 4 November 2007 pp 505-515
Extreme Points of the Convex Set of Joint Probability Distributions with Fixed Marginals
By using a quantum probabilistic approach we obtain a description of the extreme points of the convex set of all joint probability distributions on the product of two standard Borel spaces with fixed marginal distributions.
Volume 122 Issue 4 November 2012 pp 635-644
A Note on Gaussian Distributions in $\mathbb{R}^n$
B G Manjunath K R Parthasarathy
Given any finite set $\mathcal{F}$ of $(n-1)$-dimensional subspaces of $\mathbb{R}^n$ we give examples of nonGaussian probability measures in $\mathbb{R}^n$ whose marginal distribution in each subspace from $\mathcal{F}$ is Gaussian. However, if $\mathcal{F}$ is an infinite family of such $(n-1)$-dimensional subspaces then such a nonGaussian probability measure in $\mathbb{R}^n$ does not exist.
Volume 123 Issue 1 February 2013 pp 75-84
Two Remarks on Normality Preserving Borel Automorphisms of $\mathbb{R}^n$
Let 𝑇 be a bijective map on $\mathbb{R}^n$ such that both 𝑇 and $T^{-1}$ are Borel measurable. For any $\theta\in\mathbb{R}^n$ and any real $n\times n$ positive definite matrix 𝛴 , let $N(\theta,\Sigma)$ denote the 𝑛-variate normal (Gaussian) probability measure on $\mathbb{R}^n$ with mean vector 𝜃 and covariance matrix 𝛴 . Here we prove the following two results: (1) Suppose $N(\theta_j, I)T^{-1}$ is gaussian for $0\leq j\leq n$, where 𝐼 is the identity matrix and $\{\theta_j-\theta_0,1\leq j\leq n\}$ is a basis for $\mathbb{R}^n$. Then 𝑇 is an affine linear transformation; (2) Let $\Sigma_j=I+\varepsilon_ju_j{u'}_j, 1\leq j\leq n$ where $\varepsilon_j>-1$ for every 𝑗 and $\{u_j,1\leq j\leq n\}$ is a basis of unit vectors in $\mathbb{R}^n$ with ${u'}_j$ denoting the transpose of the column vector $u_j$. Suppose $N(0,I)T^{-1}$ and $N(0,\Sigma_j)T^{-1}, 1\leq j\leq n$ are gaussian. Then $T(x)=\Sigma_s 1_{E_s}(x)VsUx a.e.x$, where 𝑠 runs over the set of $2^n$ diagonal matrices of order 𝑛 with diagonal entries $\pm 1,U,V$ are $n\times n$ orthogonal matrices and $\{E_s\}$ is a collection of $2^n$ Borel subsets of $\mathbb{R}^n$ such that $\{E_s\}$ and $\{VsU(E_s)\}$ are partitions of $\mathbb{R}^n$ modulo Lebesgue-null sets and for every $j,VsU\Sigma_j(VsU)^{-1}$ is independent of all 𝑠 for which the Lebesgue measure of $E_s$ is positive. The converse of this result also holds.
Our results constitute a sharpening of the results of Nabeya and Kariya (
Volume 130, 2020
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