A theorem by Harald Cramér?Stein's method proof of the Berry-Esséen theoremStable local limit theoremsA generalization of the Sanov TheoremOn a result of Montgomery and Vaughan about Euler's totientIntuition of law of iterated logarithm?Laws of Iterated Logarithm for Random Matrices and Random PermutationUniform Law Of Iterated Logarithm for VC classesReference to iterated logarithm law and Smirnov law of empirical CDFSurely recurrent random walks and the law of the iterated logarithmIterated logarithm and gaussian concentration : a paradox
A theorem by Harald Cramér?
Stein's method proof of the Berry-Esséen theoremStable local limit theoremsA generalization of the Sanov TheoremOn a result of Montgomery and Vaughan about Euler's totientIntuition of law of iterated logarithm?Laws of Iterated Logarithm for Random Matrices and Random PermutationUniform Law Of Iterated Logarithm for VC classesReference to iterated logarithm law and Smirnov law of empirical CDFSurely recurrent random walks and the law of the iterated logarithmIterated logarithm and gaussian concentration : a paradox
$begingroup$
In the paper “On the order of magnitude of the difference between consecutive prime numbers” by Harald Cramér there is the following statement:
Suppose $X_n_n=2^infty$ is a sequence of independent random variables, such that $X_n sim Bern(frac1ln(n))$.
Then $lim_n to infty sup |fracsqrtln(n)(Sigma_i=2^n X_i - li(n))sqrt2n ln(ln(n))| = 1$
However, he does not prove this result there, but rather states, that it is proved in his paper “Prime numbers and probability” (which I could not find)
My question is:
How can this statement be proved?
Probably, it has something to do with the Law of Iterated Logarithm, but I do not know for sure ...
nt.number-theory reference-request pr.probability analytic-number-theory stochastic-processes
edited Aug 5 at 11:14
kjetil b halvorsen
1,4003 gold badges18 silver badges31 bronze badges
asked Aug 4 at 19:58
Yanior WegYanior Weg
8365 silver badges19 bronze badges
$endgroup$
add a comment |
$begingroup$
In the paper “On the order of magnitude of the difference between consecutive prime numbers” by Harald Cramér there is the following statement:
Suppose $X_n_n=2^infty$ is a sequence of independent random variables, such that $X_n sim Bern(frac1ln(n))$.
Then $lim_n to infty sup |fracsqrtln(n)(Sigma_i=2^n X_i - li(n))sqrt2n ln(ln(n))| = 1$
However, he does not prove this result there, but rather states, that it is proved in his paper “Prime numbers and probability” (which I could not find)
My question is:
How can this statement be proved?
Probably, it has something to do with the Law of Iterated Logarithm, but I do not know for sure ...
nt.number-theory reference-request pr.probability analytic-number-theory stochastic-processes
edited Aug 5 at 11:14
kjetil b halvorsen
1,4003 gold badges18 silver badges31 bronze badges
asked Aug 4 at 19:58
Yanior WegYanior Weg
8365 silver badges19 bronze badges
$endgroup$
add a comment |
$begingroup$
In the paper “On the order of magnitude of the difference between consecutive prime numbers” by Harald Cramér there is the following statement:
Suppose $X_n_n=2^infty$ is a sequence of independent random variables, such that $X_n sim Bern(frac1ln(n))$.
Then $lim_n to infty sup |fracsqrtln(n)(Sigma_i=2^n X_i - li(n))sqrt2n ln(ln(n))| = 1$
However, he does not prove this result there, but rather states, that it is proved in his paper “Prime numbers and probability” (which I could not find)
My question is:
How can this statement be proved?
Probably, it has something to do with the Law of Iterated Logarithm, but I do not know for sure ...
nt.number-theory reference-request pr.probability analytic-number-theory stochastic-processes
edited Aug 5 at 11:14
kjetil b halvorsen
1,4003 gold badges18 silver badges31 bronze badges
asked Aug 4 at 19:58
Yanior WegYanior Weg
8365 silver badges19 bronze badges
$endgroup$
In the paper “On the order of magnitude of the difference between consecutive prime numbers” by Harald Cramér there is the following statement:
Suppose $X_n_n=2^infty$ is a sequence of independent random variables, such that $X_n sim Bern(frac1ln(n))$.
Then $lim_n to infty sup |fracsqrtln(n)(Sigma_i=2^n X_i - li(n))sqrt2n ln(ln(n))| = 1$
However, he does not prove this result there, but rather states, that it is proved in his paper “Prime numbers and probability” (which I could not find)
My question is:
How can this statement be proved?
Probably, it has something to do with the Law of Iterated Logarithm, but I do not know for sure ...
nt.number-theory reference-request pr.probability analytic-number-theory stochastic-processes
nt.number-theory reference-request pr.probability analytic-number-theory stochastic-processes
edited Aug 5 at 11:14
kjetil b halvorsen
1,4003 gold badges18 silver badges31 bronze badges
asked Aug 4 at 19:58
Yanior WegYanior Weg
8365 silver badges19 bronze badges
edited Aug 5 at 11:14
kjetil b halvorsen
1,4003 gold badges18 silver badges31 bronze badges
asked Aug 4 at 19:58
Yanior WegYanior Weg
8365 silver badges19 bronze badges
edited Aug 5 at 11:14
kjetil b halvorsen
1,4003 gold badges18 silver badges31 bronze badges
edited Aug 5 at 11:14
kjetil b halvorsen
1,4003 gold badges18 silver badges31 bronze badges
edited Aug 5 at 11:14
kjetil b halvorsen
1,4003 gold badges18 silver badges31 bronze badges
1,4003 gold badges18 silver badges31 bronze badges
asked Aug 4 at 19:58
Yanior WegYanior Weg
8365 silver badges19 bronze badges
asked Aug 4 at 19:58
Yanior WegYanior Weg
8365 silver badges19 bronze badges
asked Aug 4 at 19:58
Yanior WegYanior Weg
8365 silver badges19 bronze badges
8365 silver badges19 bronze badges
add a comment |
add a comment |
1 Answer
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$begingroup$
This is a special case of a general law of the iterated logarithm for non-iid random variables (r.v.'s), which states the following:
Suppose that $Y_1,Y_2,dots$ are independent zero-mean r.v.'s, $S_n:=sum_1^n Y_i$, $B_n:=Var, S_ntoinfty$, $|Y_n|le M_nin(0,infty)$, and $M_n=o((B_n/lnln B_n)^1/2)$. Then
$$limsup_nfracS_nsqrt2B_nlnln B_n=1
$$
almost surely.
See e.g. V. Petrov, Ch. X, Theorem 1. This theorem is due to Kolmogorov (1929).
(Just in case, here is a reference to Cramér's paper:
Cramér, H. 1935 Prime numbers and probability. Skand. Mat.-Kongr. 8, 107--115.
I found it in Granville's paper.)
answered Aug 4 at 20:35
Iosif PinelisIosif Pinelis
24.3k3 gold badges26 silver badges64 bronze badges
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1 Answer
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1 Answer
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active
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$begingroup$
This is a special case of a general law of the iterated logarithm for non-iid random variables (r.v.'s), which states the following:
Suppose that $Y_1,Y_2,dots$ are independent zero-mean r.v.'s, $S_n:=sum_1^n Y_i$, $B_n:=Var, S_ntoinfty$, $|Y_n|le M_nin(0,infty)$, and $M_n=o((B_n/lnln B_n)^1/2)$. Then
$$limsup_nfracS_nsqrt2B_nlnln B_n=1
$$
almost surely.
See e.g. V. Petrov, Ch. X, Theorem 1. This theorem is due to Kolmogorov (1929).
(Just in case, here is a reference to Cramér's paper:
Cramér, H. 1935 Prime numbers and probability. Skand. Mat.-Kongr. 8, 107--115.
I found it in Granville's paper.)
answered Aug 4 at 20:35
Iosif PinelisIosif Pinelis
24.3k3 gold badges26 silver badges64 bronze badges
$endgroup$
add a comment |
$begingroup$
This is a special case of a general law of the iterated logarithm for non-iid random variables (r.v.'s), which states the following:
Suppose that $Y_1,Y_2,dots$ are independent zero-mean r.v.'s, $S_n:=sum_1^n Y_i$, $B_n:=Var, S_ntoinfty$, $|Y_n|le M_nin(0,infty)$, and $M_n=o((B_n/lnln B_n)^1/2)$. Then
$$limsup_nfracS_nsqrt2B_nlnln B_n=1
$$
almost surely.
See e.g. V. Petrov, Ch. X, Theorem 1. This theorem is due to Kolmogorov (1929).
(Just in case, here is a reference to Cramér's paper:
Cramér, H. 1935 Prime numbers and probability. Skand. Mat.-Kongr. 8, 107--115.
I found it in Granville's paper.)
answered Aug 4 at 20:35
Iosif PinelisIosif Pinelis
24.3k3 gold badges26 silver badges64 bronze badges
$endgroup$
add a comment |
$begingroup$
This is a special case of a general law of the iterated logarithm for non-iid random variables (r.v.'s), which states the following:
Suppose that $Y_1,Y_2,dots$ are independent zero-mean r.v.'s, $S_n:=sum_1^n Y_i$, $B_n:=Var, S_ntoinfty$, $|Y_n|le M_nin(0,infty)$, and $M_n=o((B_n/lnln B_n)^1/2)$. Then
$$limsup_nfracS_nsqrt2B_nlnln B_n=1
$$
almost surely.
See e.g. V. Petrov, Ch. X, Theorem 1. This theorem is due to Kolmogorov (1929).
(Just in case, here is a reference to Cramér's paper:
Cramér, H. 1935 Prime numbers and probability. Skand. Mat.-Kongr. 8, 107--115.
I found it in Granville's paper.)
answered Aug 4 at 20:35
Iosif PinelisIosif Pinelis
24.3k3 gold badges26 silver badges64 bronze badges
$endgroup$
This is a special case of a general law of the iterated logarithm for non-iid random variables (r.v.'s), which states the following:
Suppose that $Y_1,Y_2,dots$ are independent zero-mean r.v.'s, $S_n:=sum_1^n Y_i$, $B_n:=Var, S_ntoinfty$, $|Y_n|le M_nin(0,infty)$, and $M_n=o((B_n/lnln B_n)^1/2)$. Then
$$limsup_nfracS_nsqrt2B_nlnln B_n=1
$$
almost surely.
See e.g. V. Petrov, Ch. X, Theorem 1. This theorem is due to Kolmogorov (1929).
(Just in case, here is a reference to Cramér's paper:
Cramér, H. 1935 Prime numbers and probability. Skand. Mat.-Kongr. 8, 107--115.
I found it in Granville's paper.)
answered Aug 4 at 20:35
Iosif PinelisIosif Pinelis
24.3k3 gold badges26 silver badges64 bronze badges
edited Aug 4 at 21:47
answered Aug 4 at 20:35
Iosif PinelisIosif Pinelis
24.3k3 gold badges26 silver badges64 bronze badges
answered Aug 4 at 20:35
Iosif PinelisIosif Pinelis
24.3k3 gold badges26 silver badges64 bronze badges
answered Aug 4 at 20:35
Iosif PinelisIosif Pinelis
24.3k3 gold badges26 silver badges64 bronze badges
24.3k3 gold badges26 silver badges64 bronze badges
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