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Background:

The strong law of large numbers (SLLN) is a powerful result in probability, and there has been extensive literature on when the SLLN holds. However, constructing nontrivial examples for which the SLLN fails to hold seems to be (very) hard.

K.L. Chung's famous paper "The strong law of large numbers" talks about the necessary and sufficient conditions for the SLLN to hold. However, it is for independent random variabes. On the other hand, R. Lyons' paper "Strong laws of large numbers for weakly correlated random variables" is probably the best on when the SLLN holds under dependence. One of the theorems of Lyon's is quoted below:

Let $\left\{ X_{n}\right\} _{n=1}^{\infty}$ be a sequence of real-valued, zero mean random variables such that $\mathbb{E}\left[ \left\vert X_{n}\right\vert ^{2}\right] \leq1$. If $\left\vert X_{n}\right\vert \leq1$ a.s. and $$\sum\nolimits_{N=1}^{\infty}N^{-1}\mathbb{E}\left[ \left\vert N^{-1} \sum\nolimits_{n=1}^{N}X_{n}\right\vert ^{2}\right] <\infty\text{,}$$ then $\lim_{m\rightarrow\infty}N^{-1}\sum_{n=1}^{N}X_{n}=0$ a.s.

Question:

However, could anyone please give an example for which the SLLN fails to hold in the following setting? Let $\left\{ X_{n}\right\} _{n\geq1}$ be a sequence of Bernoulli random variables such that

1. $\mathbb{P}\left(X_{n}=1\right) =p_{n}$ for which $p_{i}\neq p_{j}$ whenever $i\neq j$, and $\mathbb{P}\left(X_{n}=0\right) =1-p_{n}$
2. $\lim\inf_{n\rightarrow\infty}p_{n}>0$ and $\lim\sup_{n\rightarrow\infty }p_{n}<1$
3. $X_{n}$'s are not wekly dependent in Lyon's sense (see the displayed equation above)
4. no significant proportion (compared to $m^2$ when $m$ is large) of the sequence $X_n$ for $1 \leq n \leq m$ should contain elements that are a scalar multiple of one another (this is to avoid types of examples given by AnthonyQuas)

Follow-up

Anthony and Fedor each provided an example of dependent Bernoulli's for which the SLLN fails. Their examples have special structures, i.e., $X_n = f(Y_1, \cdots, Y_{n_0}, Y_n)$ for a fixed $n_0$ and independent Bernoulli's $Y_n$, where $f$ is a linear mapping whose functional norm is bounded from above and away from $0$. It seems that the key is to introduce infinite-range dependence (which is in sharp constrast to finite-range dependence or exchangeability; see a related post Is there a McDiarmid-type inequality for sequences with a finite range of dependence? )

Is this a general technique to construct examples for the failure of SLLN for dependent Bernoulli's?

Background:

The strong law of large numbers (SLLN) is a powerful result in probability, and there has been extensive literature on when the SLLN holds. However, constructing nontrivial examples for which the SLLN fails to hold seems to be (very) hard.

K.L. Chung's famous paper "The strong law of large numbers" talks about the necessary and sufficient conditions for the SLLN to hold. However, it is for independent random variabes. On the other hand, R. Lyons' paper "Strong laws of large numbers for weakly correlated random variables" is probably the best on when the SLLN holds under dependence. One of the theorems of Lyon's is quoted below:

Let $\left\{ X_{n}\right\} _{n=1}^{\infty}$ be a sequence of real-valued, zero mean random variables such that $\mathbb{E}\left[ \left\vert X_{n}\right\vert ^{2}\right] \leq1$. If $\left\vert X_{n}\right\vert \leq1$ a.s. and $$\sum\nolimits_{N=1}^{\infty}N^{-1}\mathbb{E}\left[ \left\vert N^{-1} \sum\nolimits_{n=1}^{N}X_{n}\right\vert ^{2}\right] <\infty\text{,}$$ then $\lim_{m\rightarrow\infty}N^{-1}\sum_{n=1}^{N}X_{n}=0$ a.s.

Question:

However, could anyone please give an example for which the SLLN fails to hold in the following setting? Let $\left\{ X_{n}\right\} _{n\geq1}$ be a sequence of Bernoulli random variables such that

1. $\mathbb{P}\left(X_{n}=1\right) =p_{n}$ for which $p_{i}\neq p_{j}$ whenever $i\neq j$, and $\mathbb{P}\left(X_{n}=0\right) =1-p_{n}$
2. $\lim\inf_{n\rightarrow\infty}p_{n}>0$ and $\lim\sup_{n\rightarrow\infty }p_{n}<1$
3. $X_{n}$'s are not wekly dependent in Lyon's sense (see the displayed equation above)
4. no significant proportion (compared to $m^2$ when $m$ is large) of the sequence $X_n$ for $1 \leq n \leq m$ should contain elements that are a scalar multiple of one another (this is to avoid types of examples given by AnthonyQuas)

Follow-up

Anthony and Fedor each provided an example of dependent Bernoulli's for which the SLLN fails. Their examples have special structures, i.e., $X_n = f(Y_1, \cdots, Y_{n_0}, Y_n)$ for a fixed $n_0$ and independent Bernoulli's $Y_n$, where $f$ is a linear mapping whose functional norm is bounded from above and away from $0$. It seems that the key is to introduce infinite-range dependence (which is in sharp constrast to finite-range dependence or exchangeability; see a related post Is there a McDiarmid-type inequality for sequences with a finite range of dependence? )

Is this a general technique to construct examples for the failure of SLLN for dependent Bernoulli's?

Background:

The strong law of large numbers (SLLN) is a powerful result in probability, and there has been extensive literature on when the SLLN holds. However, constructing nontrivial examples for which the SLLN fails to hold seems to be (very) hard.

K.L. Chung's famous paper "The strong law of large numbers" talks about the necessary and sufficient conditions for the SLLN to hold. However, it is for independent random variabes. On the other hand, R. Lyons' paper "Strong laws of large numbers for weakly correlated random variables" is probably the best on when the SLLN holds under dependence. One of the theorems of Lyon's is quoted below:

Let $\left\{ X_{n}\right\} _{n=1}^{\infty}$ be a sequence of real-valued, zero mean random variables such that $\mathbb{E}\left[ \left\vert X_{n}\right\vert ^{2}\right] \leq1$. If $\left\vert X_{n}\right\vert \leq1$ a.s. and $$\sum\nolimits_{N=1}^{\infty}N^{-1}\mathbb{E}\left[ \left\vert N^{-1} \sum\nolimits_{n=1}^{N}X_{n}\right\vert ^{2}\right] <\infty\text{,}$$ then $\lim_{m\rightarrow\infty}N^{-1}\sum_{n=1}^{N}X_{n}=0$ a.s.

Question:

However, could anyone please give an example for which the SLLN fails to hold in the following setting? Let $\left\{ X_{n}\right\} _{n\geq1}$ be a sequence of Bernoulli random variables such that

1. $\mathbb{P}\left(X_{n}=1\right) =p_{n}$ for which $p_{i}\neq p_{j}$ whenever $i\neq j$, and $\mathbb{P}\left(X_{n}=0\right) =1-p_{n}$
2. $\lim\inf_{n\rightarrow\infty}p_{n}>0$ and $\lim\sup_{n\rightarrow\infty }p_{n}<1$
3. $X_{n}$'s are not wekly dependent in Lyon's sense (see the displayed equation above)
4. no significant proportion (compared to $m^2$ when $m$ is large) of the sequence $X_n$ for $1 \leq n \leq m$ should contain elements that are a scalar multiple of one another (this is to avoid types of examples given by AnthonyQuas)

Follow-up

Anthony and Fedor each provided an example of dependent Bernoulli's for which the SLLN fails. Their examples have special structures, i.e., $X_n = f(Y_1, \cdots, Y_{n_0}, Y_n)$ for a fixed $n_0$ and independent Bernoulli's $Y_n$, where $f$ is a linear mapping whose functional norm is bounded from above and away from $0$.

Is this a general technique to construct examples for the failure of SLLN for dependent Bernoulli's?

Background:

The strong law of large numbers (SLLN) is a powerful result in probability, and there has been extensive literature on when the SLLN holds. However, constructing nontrivial examples for which the SLLN fails to hold seems to be (very) hard.

K.L. Chung's famous paper "The strong law of large numbers" talks about the necessary and sufficient conditions for the SLLN to hold. However, it is for independent random variabes. On the other hand, R. Lyons' paper "Strong laws of large numbers for weakly correlated random variables" is probably the best on when the SLLN holds under dependence. One of the theorems of Lyon's is quoted below:

Let $\left\{ X_{n}\right\} _{n=1}^{\infty}$ be a sequence of real-valued, zero mean random variables such that $\mathbb{E}\left[ \left\vert X_{n}\right\vert ^{2}\right] \leq1$. If $\left\vert X_{n}\right\vert \leq1$ a.s. and $$\sum\nolimits_{N=1}^{\infty}N^{-1}\mathbb{E}\left[ \left\vert N^{-1} \sum\nolimits_{n=1}^{N}X_{n}\right\vert ^{2}\right] <\infty\text{,}$$ then $\lim_{m\rightarrow\infty}N^{-1}\sum_{n=1}^{N}X_{n}=0$ a.s.

Question:

However, could anyone please give an example for which the SLLN fails to hold in the following setting? Let $\left\{ X_{n}\right\} _{n\geq1}$ be a sequence of Bernoulli random variables such that

1. $\mathbb{P}\left(X_{n}=1\right) =p_{n}$ for which $p_{i}\neq p_{j}$ whenever $i\neq j$, and $\mathbb{P}\left(X_{n}=0\right) =1-p_{n}$
2. $\lim\inf_{n\rightarrow\infty}p_{n}>0$ and $\lim\sup_{n\rightarrow\infty }p_{n}<1$
3. $X_{n}$'s are not wekly dependent in Lyon's sense (see the displayed equation above)
4. no significant proportion (compared to $m^2$ when $m$ is large) of the sequence $X_n$ for $1 \leq n \leq m$ should contain elements that are a scalar multiple of one another (this is to avoid types of examples given by AnthonyQuas)

Follow-up

Anthony and Fedor each provided an example of dependent Bernoulli's for which the SLLN fails. Their examples have special structures, i.e., $X_n = f(Y_1, \cdots, Y_{n_0}, Y_n)$ for a fixed $n_0$ and independent Bernoulli's $Y_n$, where $f$ is a linear mapping whose functional norm is bounded from above and away from $0$. It seems that the key is to introduce infinite-range dependence (which is in sharp constrast to finite-range dependence or exchangeability; see a related post Is there a McDiarmid-type inequality for sequences with a finite range of dependence? )

Is this a general technique to construct examples for the failure of SLLN for dependent Bernoulli's?

Background:

The strong law of large numbers (SLLN) is a powerful result in probability, and there has been extensive literature on when the SLLN holds. However, constructing nontrivial examples for which the SLLN fails to hold seems to be (very) hard.

K.L. Chung's famous paper "The strong law of large numbers" talks about the necessary and sufficient conditions for the SLLN to hold. However, it is for independent random variabes. On the other hand, R. Lyons' paper "Strong laws of large numbers for weakly correlated random variables" is probably the best on when the SLLN holds under dependence. One of the theorems of Lyon's is quoted below:

Let $\left\{ X_{n}\right\} _{n=1}^{\infty}$ be a sequence of real-valued, zero mean random variables such that $\mathbb{E}\left[ \left\vert X_{n}\right\vert ^{2}\right] \leq1$. If $\left\vert X_{n}\right\vert \leq1$ a.s. and $$\sum\nolimits_{N=1}^{\infty}N^{-1}\mathbb{E}\left[ \left\vert N^{-1} \sum\nolimits_{n=1}^{N}X_{n}\right\vert ^{2}\right] <\infty\text{,}$$ then $\lim_{m\rightarrow\infty}N^{-1}\sum_{n=1}^{N}X_{n}=0$ a.s.

Question:

However, could anyone please give an example for which the SLLN fails to hold in the following setting? Let $\left\{ X_{n}\right\} _{n\geq1}$ be a sequence of Bernoulli random variables such that

1. $\mathbb{P}\left(X_{n}=1\right) =p_{n}$ for which $p_{i}\neq p_{j}$ whenever $i\neq j$, and $\mathbb{P}\left(X_{n}=0\right) =1-p_{n}$
2. $\lim\inf_{n\rightarrow\infty}p_{n}>0$ and $\lim\sup_{n\rightarrow\infty }p_{n}<1$
3. $X_{n}$'s are not wekly dependent in Lyon's sense (see the displayed equation above)
4. no significant proportion (compared to $m^2$ when $m$ is large) of the sequence $X_n$ for $1 \leq n \leq m$ should contain elements that are a scalar multiple of one another (this is to avoid types of examples given by AnthonyQuas)

Background:

The strong law of large numbers (SLLN) is a powerful result in probability, and there has been extensive literature on when the SLLN holds. However, constructing nontrivial examples for which the SLLN fails to hold seems to be (very) hard.

K.L. Chung's famous paper "The strong law of large numbers" talks about the necessary and sufficient conditions for the SLLN to hold. However, it is for independent random variabes. On the other hand, R. Lyons' paper "Strong laws of large numbers for weakly correlated random variables" is probably the best on when the SLLN holds under dependence. One of the theorems of Lyon's is quoted below:

Let $\left\{ X_{n}\right\} _{n=1}^{\infty}$ be a sequence of real-valued, zero mean random variables such that $\mathbb{E}\left[ \left\vert X_{n}\right\vert ^{2}\right] \leq1$. If $\left\vert X_{n}\right\vert \leq1$ a.s. and $$\sum\nolimits_{N=1}^{\infty}N^{-1}\mathbb{E}\left[ \left\vert N^{-1} \sum\nolimits_{n=1}^{N}X_{n}\right\vert ^{2}\right] <\infty\text{,}$$ then $\lim_{m\rightarrow\infty}N^{-1}\sum_{n=1}^{N}X_{n}=0$ a.s.

Question:

However, could anyone please give an example for which the SLLN fails to hold in the following setting? Let $\left\{ X_{n}\right\} _{n\geq1}$ be a sequence of Bernoulli random variables such that

1. $\mathbb{P}\left(X_{n}=1\right) =p_{n}$ for which $p_{i}\neq p_{j}$ whenever $i\neq j$, and $\mathbb{P}\left(X_{n}=0\right) =1-p_{n}$
2. $\lim\inf_{n\rightarrow\infty}p_{n}>0$ and $\lim\sup_{n\rightarrow\infty }p_{n}<1$
3. $X_{n}$'s are not wekly dependent in Lyon's sense (see the displayed equation above)
4. no significant proportion (compared to $m^2$ when $m$ is large) of the sequence $X_n$ for $1 \leq n \leq m$ should contain elements that are a scalar multiple of one another (this is to avoid types of examples given by AnthonyQuas)

Follow-up

Anthony and Fedor each provided an example of dependent Bernoulli's for which the SLLN fails. Their examples have special structures, i.e., $X_n = f(Y_1, \cdots, Y_{n_0}, Y_n)$ for a fixed $n_0$ and independent Bernoulli's $Y_n$, where $f$ is a linear mapping whose functional norm is bounded from above and away from $0$.

Is this a general technique to construct examples for the failure of SLLN for dependent Bernoulli's?