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If we assume that the $X_i$ are iid in $L^p$ for some $p\in ]1,2[$, then we have:

$$ a.e. \ \ {1 \over n} \sum_{k=0}^{n−1} X_k=E(X_0)+o(n^{1/p−1})$$

This follows from the Kolmogorov three series theorem. This is done in the book of Durrett, probability, theory and examples, theorem 2.5.8.

Note that if you are not interested by the exact exponent, then the standard quick $L^2$ proof gives you such an estimate.

If we assume that the $X_i$ are iid in $L^p$ for some $p\in ]1,2[$, then we have:

$$ a.e. \ \ {1 \over n} \sum_{k=0}^{n−1} X_k=E(X_0)+o(n^{1/p−1})$$

This follows from the Kolmogorov three series theorem. This is done in the book of Durrett, probability, theory and examples, theorem 2.5.8 (now theorem 2.5.12 with the 5th version of the book).

Note that if you are not interested by the exact exponent, then the standard quick $L^2$ proof gives you such an estimate.

If we assume that the $X_i$ are iid in $L^p$ for some $p\in ]1,2[$, then we have:

$$ a.e. \ \ {1 \over n} \sum_{k=0}^{n−1} X_k=E(X_0)+o(n^{1/p−1})$$

This is a standard exercice. Use the fact that if $Y_i$ is centered, independent such that $\sum Var(Y_i)$ is convergent, then the series $\sum Y_i$ converges a.e. Then take $Y_i = X_i/i^\alpha$ and conclude with the Kronecker lemma. This is a standard lemma used in the proof of the Kolmogorov three series theorem. This is done in the book of Durrett, probability, theory and examples. theorem 2.5.8.

Note that if you are not interested by the exact exponent, then the standard quick $L^2$ proof gives you such an estimate.

If we assume that the $X_i$ are iid in $L^p$ for some $p\in ]1,2[$, then we have:

$$ a.e. \ \ {1 \over n} \sum_{k=0}^{n−1} X_k=E(X_0)+o(n^{1/p−1})$$

This follows from the Kolmogorov three series theorem. This is done in the book of Durrett, probability, theory and examples, theorem 2.5.8.

Note that if you are not interested by the exact exponent, then the standard quick $L^2$ proof gives you such an estimate.

If we assume that the $X_i$ are iid in $L^p$ for some $p\in ]1,2[$, then we have:

$$ a.e. \ \ {1 \over n} \sum_{k=0}^{n−1} X_k=E(X_0)+o(n^{1/p−1})$$

This is a standard exercice. Use the fact that if $Y_i$ is centered, independent such that $\sum Var(Y_i)$ is convergent, then the series $\sum Y_i$ converges a.e. Then take $Y_i = X_i/i^\alpha$ and conclude with the Kronecker lemma. This is a standard lemma used in the proof of the Kolmogorov three series theorem. This is probably done in the book of Durrett, probability, theory and examples.

Note that if you are not interested by the exact exponent, then the standard quick $L^2$ proof gives you such an estimate.

If we assume that the $X_i$ are iid in $L^p$ for some $p\in ]1,2[$, then we have:

$$ a.e. \ \ {1 \over n} \sum_{k=0}^{n−1} X_k=E(X_0)+o(n^{1/p−1})$$

This is a standard exercice. Use the fact that if $Y_i$ is centered, independent such that $\sum Var(Y_i)$ is convergent, then the series $\sum Y_i$ converges a.e. Then take $Y_i = X_i/i^\alpha$ and conclude with the Kronecker lemma. This is a standard lemma used in the proof of the Kolmogorov three series theorem. This is done in the book of Durrett, probability, theory and examples. theorem 2.5.8.

Note that if you are not interested by the exact exponent, then the standard quick $L^2$ proof gives you such an estimate.