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?
$$
limsup_{ntoinfty} frac{X_1+cdots+X_n}{n^{alpha}(loglog n)^{1-alpha}}=1,quad text{a.s.}
$$
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
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Ali Khezeli
41327
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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?
$$
limsup_{ntoinfty} frac{X_1+cdots+X_n}{n^{alpha}(loglog n)^{1-alpha}}=1,quad text{a.s.}
$$
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 yesterday
Ali Khezeli
41327
add a comment |
up vote
1
down vote
favorite
up vote
1
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?
$$
limsup_{ntoinfty} frac{X_1+cdots+X_n}{n^{alpha}(loglog n)^{1-alpha}}=1,quad text{a.s.}
$$
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 yesterday
Ali Khezeli
41327
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?
$$
limsup_{ntoinfty} frac{X_1+cdots+X_n}{n^{alpha}(loglog n)^{1-alpha}}=1,quad text{a.s.}
$$
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 yesterday
Ali Khezeli
41327
asked yesterday
Ali Khezeli
41327
asked yesterday
Ali Khezeli
41327
asked yesterday
Ali Khezeli
41327
asked yesterday
Ali Khezeli
41327
41327
add a comment |
add a comment |
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