Law of Iterated Logarithm when the mean does not exist
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Let $X_1,X_2,ldots$ be an i.i.d. sequence of random variables such that $X_1geq 0$ a.s. and $mathbb P[X_1>x]sim x^{-alpha}$, where $alpha<2$. This implies that $X_1$ does not have finite mean. Does the following law of iterated logarithm hold?
$$
limsup_{ntoinfty} frac{X_1+cdots+X_n}{n^{alpha}(loglog n)^{1-alpha}}=c,quad text{a.s.},
$$
where $c$ is a constant depending on the distribution of $X_1$.
A continuous-time version holds (see here). Also, a special case of the claim exists here (Theorem 3) regarding zeros of the simple random walk, in which $alpha=frac 12$.
probability-theory probability-limit-theorems
asked Nov 14 at 5:54
Ali Khezeli
37328
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Let $X_1,X_2,ldots$ be an i.i.d. sequence of random variables such that $X_1geq 0$ a.s. and $mathbb P[X_1>x]sim x^{-alpha}$, where $alpha<2$. This implies that $X_1$ does not have finite mean. Does the following law of iterated logarithm hold?
$$
limsup_{ntoinfty} frac{X_1+cdots+X_n}{n^{alpha}(loglog n)^{1-alpha}}=c,quad text{a.s.},
$$
where $c$ is a constant depending on the distribution of $X_1$.
A continuous-time version holds (see here). Also, a special case of the claim exists here (Theorem 3) regarding zeros of the simple random walk, in which $alpha=frac 12$.
probability-theory probability-limit-theorems
asked Nov 14 at 5:54
Ali Khezeli
37328
This question has an open bounty worth +50
reputation from Ali Khezeli ending in 2 days.
Looking for an answer drawing from credible and/or official sources.
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Let $X_1,X_2,ldots$ be an i.i.d. sequence of random variables such that $X_1geq 0$ a.s. and $mathbb P[X_1>x]sim x^{-alpha}$, where $alpha<2$. This implies that $X_1$ does not have finite mean. Does the following law of iterated logarithm hold?
$$
limsup_{ntoinfty} frac{X_1+cdots+X_n}{n^{alpha}(loglog n)^{1-alpha}}=c,quad text{a.s.},
$$
where $c$ is a constant depending on the distribution of $X_1$.
A continuous-time version holds (see here). Also, a special case of the claim exists here (Theorem 3) regarding zeros of the simple random walk, in which $alpha=frac 12$.
probability-theory probability-limit-theorems
asked Nov 14 at 5:54
Ali Khezeli
37328
Let $X_1,X_2,ldots$ be an i.i.d. sequence of random variables such that $X_1geq 0$ a.s. and $mathbb P[X_1>x]sim x^{-alpha}$, where $alpha<2$. This implies that $X_1$ does not have finite mean. Does the following law of iterated logarithm hold?
$$
limsup_{ntoinfty} frac{X_1+cdots+X_n}{n^{alpha}(loglog n)^{1-alpha}}=c,quad text{a.s.},
$$
where $c$ is a constant depending on the distribution of $X_1$.
A continuous-time version holds (see here). Also, a special case of the claim exists here (Theorem 3) regarding zeros of the simple random walk, in which $alpha=frac 12$.
probability-theory probability-limit-theorems
probability-theory probability-limit-theorems
asked Nov 14 at 5:54
Ali Khezeli
37328
asked Nov 14 at 5:54
Ali Khezeli
37328
edited Nov 17 at 6:55
asked Nov 14 at 5:54
Ali Khezeli
37328
asked Nov 14 at 5:54
Ali Khezeli
37328
asked Nov 14 at 5:54
Ali Khezeli
37328
37328
This question has an open bounty worth +50
reputation from Ali Khezeli ending in 2 days.
Looking for an answer drawing from credible and/or official sources.
This question has an open bounty worth +50
reputation from Ali Khezeli ending in 2 days.
Looking for an answer drawing from credible and/or official sources.
add a comment |
add a comment |
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