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

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 $\ell^1$. 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 $\ell^2$.Comment: 10 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 $X$ 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