229 research outputs found
A variational formula on the Cram\'er function of series of independent random variables
In [11] it has been proved some variational formula on the Legendre-Fenchel
transform of the cumulant generating function (the Cram\'er function) of
Rademacher series with coefficients in the space . In this paper we
show a generalization of this formula to series of a larger class of any
independent random variables with coefficients that belong to the space
.Comment: 10 page
A note on the gambling team method
Gerber and Li in \cite{GeLi} formulated, using a Markov chain embedding, a
system of equations that describes relations between generating functions of
waiting time distributions for occurrences of patterns in a sequence of
independent repeated experiments when initial outcomes of the process are
known. We show how this system of equations can be obtained by using the
classical gambling team technique . We also present a form of solution of the
system and give an example showing how first results of trials influence the
probabilities that a chosen pattern precedes remaining ones in a realization of
the process.Comment: 9 page
Penney's game between many players
We recall a combinatorial derivation of the functions generating probability
of winnings for each of many participants of the Penney's game and show a
generalization of the Conway's formula to this case.Comment: 6 page
Cram\'er transform and t-entropy
t-entropy is the convex conjugate of the logarithm of the spectral radius of
a weighted composition operator (WCO). Let be a nonnegative random
variable. We show how the Cram\'er transform with respect to the spectral
radius of WCO is expressed by the t-entropy and the Cram\'er transform of the
given random variable X.Comment: 12 pages; Positivity(2013
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