I am an engineering student, new to this forum, and hope I can benefit from your knowledge. I am no mathematician and am really stuck with this problem. I hope someone can help me out.
Logconcavity of probabilities is often defined based on the probability density function. I would like to define it based on the cumulative distribution function (CDF). So a distribution is logconcave if its CDF is logconcave. Now, let's extend this to a multivariate setting.
If X is an m-variate distribution with a logconcave CDF, I would like to know if the linear projection of this vector into a 1-dimensional space i.e. a(Transpose)X which is a univariate distribution has a logconcave CDF as well.
I would actually like to use this as a trick to use a univariate Chebychev-like inequality that we have developped for distributions with a logconcave CDF (e.g. Lognormal distribution), but as a projection of a higher-order probability.
Many thanks!