Why is theorem 10.2 not applicable to the sequence of 'winnings' (left{X_{1}, X_{2}, ldots ight}), which arises

Why is theorem 10.2 not applicable to the sequence of 'winnings' \(\left\{X_{1}, X_{2}, \ldots\right\}\), which arises by applying the doubling strategy (example 10.6)?

Data from Example 10.6

Data from Theorem 10.2

Transcribed Image Text:

Example 10.6 The structure of X given by (10.14) includes as a special case the winnings (losses) development when applying the doubling strategy: Jean bets 1 on the first game. If he wins, he gets 1. If he loses, his 'winning' are - 1. Hoping to equalize the loss, Jean will bet 2 on the next game. If he wins, he will get 2 and, hence, will have made total winnings of 1. But if he loses he will have total 'winn- ings' of -3, and will bet 4 on the next game and so on. After the first win Jean stops gambling. The following table shows the losses (winnings) development of Jean if he loses 5 times in a row and then wins: game 1 2 3 4 5 6 result loss loss loss loss loss win bet 1 2 4 8 16 32 'total winnings' -1 -3 -7 -15 -31 +1