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On the Asymptotic Behaviour of a Function Arising from Tossing Coins

On the Asymptotic Behaviour of a Function Arising from Tossing Coins ON THE ASYMPTOTIC BEHAVIOUR OF A FUNCTION ARISING FROM TOSSING COINS C. F. WOODCOCK 1. Introduction Let P,xeR with P > 1, x > 0. We pu t F(P, x) = £ [1 -( 1 -P-") ] (clearly convergent). fc-0 Then F(P, x) has the following elementary probabilistic interpretation. Suppose that N people each toss a coin until a head occurs, and the probability of each coin coming down tails is \/P. Then the expected length of the longest of the N 'coin tossings' is F(P,N). We consider the asymptotic behaviour of F(P,x) as *- • oo. It is clear from a number of points of view that F(P, x) grows roughly like log (jc) but, as we shall see, there is also an oscillatory overlay which damps down extremely rapidly as P -*• 1 (see [1] for a similar phenomenon). In fact, by applying the 'integral test', we see immediately that ('change the variable to 1—P"*'). 2. The main Theorem We shall prove the following result. THEOREM . Let 0eR with 0^0 ^ 1, and suppose that JC->OO with log (x) - \\og (x)} - 6. Then P P I 2nim \ In particular, log(P) 2 <- log(i>) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

On the Asymptotic Behaviour of a Function Arising from Tossing Coins

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/28.1.19
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Abstract

ON THE ASYMPTOTIC BEHAVIOUR OF A FUNCTION ARISING FROM TOSSING COINS C. F. WOODCOCK 1. Introduction Let P,xeR with P > 1, x > 0. We pu t F(P, x) = £ [1 -( 1 -P-") ] (clearly convergent). fc-0 Then F(P, x) has the following elementary probabilistic interpretation. Suppose that N people each toss a coin until a head occurs, and the probability of each coin coming down tails is \/P. Then the expected length of the longest of the N 'coin tossings' is F(P,N). We consider the asymptotic behaviour of F(P,x) as *- • oo. It is clear from a number of points of view that F(P, x) grows roughly like log (jc) but, as we shall see, there is also an oscillatory overlay which damps down extremely rapidly as P -*• 1 (see [1] for a similar phenomenon). In fact, by applying the 'integral test', we see immediately that ('change the variable to 1—P"*'). 2. The main Theorem We shall prove the following result. THEOREM . Let 0eR with 0^0 ^ 1, and suppose that JC->OO with log (x) - \\og (x)} - 6. Then P P I 2nim \ In particular, log(P) 2 <- log(i>)

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jan 1, 1996

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