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$>\to <$: in Spitzer's formula, $N=\inf\{n\ge 1:S_n\le 0\}$.
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zhoraster
zhoraster
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You are looking for the "mean ladder height" of a random walk. There is a (not very tractable) formula due to Spitzer that gives the answer:

$$ E(S_N)={\sigma\over\sqrt{2}} \exp\left\(\sum_{n=1}^\infty {1\over n}(P(S_n>0)-1/2)\right\)$$$$ E(S_N)={\sigma\over\sqrt{2}} \exp\left\(\sum_{n=1}^\infty {1\over n}(P(S_n<0)-1/2)\right\)$$

Here $\sigma^2$ is the variance of the jump distribution. Maybe it would be possible to work this out in your special case.

[1] Chow, Yuan S. On Spitzer's formula for the moment of ladder variables. Statist. Sinica 7 no. 1, 1997, 149–156.

[2] Spitzer, Frank A Tauberian theorem and its probability interpretation. Trans. Amer. Math. Soc. 94, 1960, 150–169.

You are looking for the "mean ladder height" of a random walk. There is a (not very tractable) formula due to Spitzer that gives the answer:

$$ E(S_N)={\sigma\over\sqrt{2}} \exp\left\(\sum_{n=1}^\infty {1\over n}(P(S_n>0)-1/2)\right\)$$

Here $\sigma^2$ is the variance of the jump distribution. Maybe it would be possible to work this out in your special case.

[1] Chow, Yuan S. On Spitzer's formula for the moment of ladder variables. Statist. Sinica 7 no. 1, 1997, 149–156.

[2] Spitzer, Frank A Tauberian theorem and its probability interpretation. Trans. Amer. Math. Soc. 94, 1960, 150–169.

You are looking for the "mean ladder height" of a random walk. There is a (not very tractable) formula due to Spitzer that gives the answer:

$$ E(S_N)={\sigma\over\sqrt{2}} \exp\left\(\sum_{n=1}^\infty {1\over n}(P(S_n<0)-1/2)\right\)$$

Here $\sigma^2$ is the variance of the jump distribution. Maybe it would be possible to work this out in your special case.

[1] Chow, Yuan S. On Spitzer's formula for the moment of ladder variables. Statist. Sinica 7 no. 1, 1997, 149–156.

[2] Spitzer, Frank A Tauberian theorem and its probability interpretation. Trans. Amer. Math. Soc. 94, 1960, 150–169.

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user6096
user6096

You are looking for the "mean ladder height" of a random walk. There is a (not very tractable) formula due to Spitzer that gives the answer:

$$ E(S_N)={\sigma\over\sqrt{2}} \exp\left\({1\over n}\sum_{n=1}^\infty (P(S_n>0)-1/2)\right\)$$$$ E(S_N)={\sigma\over\sqrt{2}} \exp\left\(\sum_{n=1}^\infty {1\over n}(P(S_n>0)-1/2)\right\)$$

Here $\sigma^2$ is the variance of the jump distribution. Maybe it would be possible to work this out in your special case.

[1] Chow, Yuan S. On Spitzer's formula for the moment of ladder variables. Statist. Sinica 7 no. 1, 1997, 149–156.

[2] Spitzer, Frank A Tauberian theorem and its probability interpretation. Trans. Amer. Math. Soc. 94, 1960, 150–169.

You are looking for the "mean ladder height" of a random walk. There is a (not very tractable) formula due to Spitzer that gives the answer:

$$ E(S_N)={\sigma\over\sqrt{2}} \exp\left\({1\over n}\sum_{n=1}^\infty (P(S_n>0)-1/2)\right\)$$

Here $\sigma^2$ is the variance of the jump distribution. Maybe it would be possible to work this out in your special case.

[1] Chow, Yuan S. On Spitzer's formula for the moment of ladder variables. Statist. Sinica 7 no. 1, 1997, 149–156.

[2] Spitzer, Frank A Tauberian theorem and its probability interpretation. Trans. Amer. Math. Soc. 94, 1960, 150–169.

You are looking for the "mean ladder height" of a random walk. There is a (not very tractable) formula due to Spitzer that gives the answer:

$$ E(S_N)={\sigma\over\sqrt{2}} \exp\left\(\sum_{n=1}^\infty {1\over n}(P(S_n>0)-1/2)\right\)$$

Here $\sigma^2$ is the variance of the jump distribution. Maybe it would be possible to work this out in your special case.

[1] Chow, Yuan S. On Spitzer's formula for the moment of ladder variables. Statist. Sinica 7 no. 1, 1997, 149–156.

[2] Spitzer, Frank A Tauberian theorem and its probability interpretation. Trans. Amer. Math. Soc. 94, 1960, 150–169.

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user6096
user6096

You are looking for the "mean ladder height" of a random walk. There is a (not very tractable) formula due to Spitzer that gives the answer:

$$ E(S_N)={\sigma\over\sqrt{2}} \exp\left\({1\over n}\sum_{n=1}^\infty (P(S_n>0)-1/2)\right\)$$

Here $\sigma^2$ is the variance of the jump distribution. Maybe it would be possible to work this out in your special case.

[1] Chow, Yuan S. On Spitzer's formula for the moment of ladder variables. Statist. Sinica 7 no. 1, 1997, 149–156.

[2] Spitzer, Frank A Tauberian theorem and its probability interpretation. Trans. Amer. Math. Soc. 94, 1960, 150–169.