Non convergence of a series of random variables
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Question:
Let $(X_n), ninmathbb{N}$ be a sequence of independent r.v.s such that $P(X_n=n^4)=frac{1}{n^4}$ and $P(X_n=-1)=1-frac{1}{n^4}$. Study the a.s. convergence of $S_n=sum_{i=1}^n X_n$ as $nrightarrow +infty$.
My Attempt:
I have been simulating the stochastic process $S_n$ in R to understand whether convergence was possible at all but in none of the simulations I performed I obtained a finite value (all of the paths go to -9999).
Also, clearly, $sum_{i=1}^n operatorname{Var}(S_n)rightarrow +infty$ as $nrightarrow +infty$. Can I thus conclude that $S_n$ does NOT converge a.s.?
Many thanks in advance for the help!
probability probability-theory convergence borel-cantelli-lemmas
edited Dec 8 at 19:42
Davide Giraudo
124k16150259
asked Dec 4 at 16:22
Giulio
273
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up vote
4
down vote
favorite
Question:
Let $(X_n), ninmathbb{N}$ be a sequence of independent r.v.s such that $P(X_n=n^4)=frac{1}{n^4}$ and $P(X_n=-1)=1-frac{1}{n^4}$. Study the a.s. convergence of $S_n=sum_{i=1}^n X_n$ as $nrightarrow +infty$.
My Attempt:
I have been simulating the stochastic process $S_n$ in R to understand whether convergence was possible at all but in none of the simulations I performed I obtained a finite value (all of the paths go to -9999).
Also, clearly, $sum_{i=1}^n operatorname{Var}(S_n)rightarrow +infty$ as $nrightarrow +infty$. Can I thus conclude that $S_n$ does NOT converge a.s.?
Many thanks in advance for the help!
probability probability-theory convergence borel-cantelli-lemmas
edited Dec 8 at 19:42
Davide Giraudo
124k16150259
asked Dec 4 at 16:22
Giulio
273
add a comment |
up vote
4
down vote
favorite
up vote
4
down vote
favorite
Question:
Let $(X_n), ninmathbb{N}$ be a sequence of independent r.v.s such that $P(X_n=n^4)=frac{1}{n^4}$ and $P(X_n=-1)=1-frac{1}{n^4}$. Study the a.s. convergence of $S_n=sum_{i=1}^n X_n$ as $nrightarrow +infty$.
My Attempt:
I have been simulating the stochastic process $S_n$ in R to understand whether convergence was possible at all but in none of the simulations I performed I obtained a finite value (all of the paths go to -9999).
Also, clearly, $sum_{i=1}^n operatorname{Var}(S_n)rightarrow +infty$ as $nrightarrow +infty$. Can I thus conclude that $S_n$ does NOT converge a.s.?
Many thanks in advance for the help!
probability probability-theory convergence borel-cantelli-lemmas
edited Dec 8 at 19:42
Davide Giraudo
124k16150259
asked Dec 4 at 16:22
Giulio
273
Question:
Let $(X_n), ninmathbb{N}$ be a sequence of independent r.v.s such that $P(X_n=n^4)=frac{1}{n^4}$ and $P(X_n=-1)=1-frac{1}{n^4}$. Study the a.s. convergence of $S_n=sum_{i=1}^n X_n$ as $nrightarrow +infty$.
My Attempt:
I have been simulating the stochastic process $S_n$ in R to understand whether convergence was possible at all but in none of the simulations I performed I obtained a finite value (all of the paths go to -9999).
Also, clearly, $sum_{i=1}^n operatorname{Var}(S_n)rightarrow +infty$ as $nrightarrow +infty$. Can I thus conclude that $S_n$ does NOT converge a.s.?
Many thanks in advance for the help!
probability probability-theory convergence borel-cantelli-lemmas
probability probability-theory convergence borel-cantelli-lemmas
edited Dec 8 at 19:42
Davide Giraudo
124k16150259
asked Dec 4 at 16:22
Giulio
273
edited Dec 8 at 19:42
Davide Giraudo
124k16150259
asked Dec 4 at 16:22
Giulio
273
edited Dec 8 at 19:42
Davide Giraudo
124k16150259
edited Dec 8 at 19:42
Davide Giraudo
124k16150259
edited Dec 8 at 19:42
Davide Giraudo
124k16150259
124k16150259
asked Dec 4 at 16:22
Giulio
273
asked Dec 4 at 16:22
Giulio
273
asked Dec 4 at 16:22
Giulio
273
273
add a comment |
add a comment |
1 Answer
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Note that $P(X_n>1quad text{i.o})=0$ by Borel cantelli. Hence $X_n=-1$ for all but finitely many $n$ with probability $1$. In particular, the series diverges.
answered Dec 4 at 16:31
Foobaz John
20.4k41250
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
Note that $P(X_n>1quad text{i.o})=0$ by Borel cantelli. Hence $X_n=-1$ for all but finitely many $n$ with probability $1$. In particular, the series diverges.
answered Dec 4 at 16:31
Foobaz John
20.4k41250
add a comment |
up vote
3
down vote
accepted
Note that $P(X_n>1quad text{i.o})=0$ by Borel cantelli. Hence $X_n=-1$ for all but finitely many $n$ with probability $1$. In particular, the series diverges.
answered Dec 4 at 16:31
Foobaz John
20.4k41250
add a comment |
up vote
3
down vote
accepted
up vote
3
down vote
accepted
Note that $P(X_n>1quad text{i.o})=0$ by Borel cantelli. Hence $X_n=-1$ for all but finitely many $n$ with probability $1$. In particular, the series diverges.
answered Dec 4 at 16:31
Foobaz John
20.4k41250
Note that $P(X_n>1quad text{i.o})=0$ by Borel cantelli. Hence $X_n=-1$ for all but finitely many $n$ with probability $1$. In particular, the series diverges.
answered Dec 4 at 16:31
Foobaz John
20.4k41250
answered Dec 4 at 16:31
Foobaz John
20.4k41250
answered Dec 4 at 16:31
Foobaz John
20.4k41250
answered Dec 4 at 16:31
Foobaz John
20.4k41250
20.4k41250
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