Probabilities of competing binomial random variables

Suppose you and your friend both do $n$ tosses of an unfair coin with probability of heads equal to $α$. What is the behavior of the probability that you obtain at least $d$ more heads than your friend if you make $r$ additional tosses? We obtain asymptotic and monotonicity/convexity properties for this competing probability as a function of $n$, and demonstrate surprising phase transition phenomenons as parameters $d, r$ and $α$ vary. Our main tools are integral representations based on Fourier analysis.

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