Convergence in probability of sum of bounded random variables implies finite expectation
1
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Let $(X_i)$ be sequence of independent random variables with $|X_i| < 1$ almost-surely and define $M_n :=sum_{i=1}^{n}X_i$. Suppose it is known that $M_n$ converges in probability to some random variable $M$, where $M < infty$ almost-surely. Does it follow that $sum_{i=1}^{infty}mathbb{E}[X_i] < infty$?
probability probability-theory convergence random-variables
asked Dec 11 '18 at 23:27
jesterIIjesterII
1,21321326
$endgroup$
add a comment |
1
$begingroup$
Let $(X_i)$ be sequence of independent random variables with $|X_i| < 1$ almost-surely and define $M_n :=sum_{i=1}^{n}X_i$. Suppose it is known that $M_n$ converges in probability to some random variable $M$, where $M < infty$ almost-surely. Does it follow that $sum_{i=1}^{infty}mathbb{E}[X_i] < infty$?
probability probability-theory convergence random-variables
asked Dec 11 '18 at 23:27
jesterIIjesterII
1,21321326
$endgroup$
$begingroup$
Yes, the series converges... the proof which i know is somewhat technical; it uses an symmetrization argument, i.e. $Y_i := X_i-tilde{X}_i$ where $tilde{X}_i$ is an independent copy of $X_i$
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– saz
Dec 12 '18 at 6:36
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@saz Any reference or further details would be appreciated.
$endgroup$
– jesterII
Dec 12 '18 at 13:30
1
$begingroup$
Do you know Kolmogorov's three series convergence theorem? You can deduce it immediately from this result.
$endgroup$
– saz
Dec 12 '18 at 13:54
$begingroup$
@saz I've come across the three series theorem, although that applies to convergence a.s. whereas in this case the series is assumed to converge in probability. Is there an immediate way to connect the two?
$endgroup$
– jesterII
Dec 12 '18 at 15:56
1
$begingroup$
A well-known result (by Lévy) states that for independent random variables $X_i$ the series $sum_i X_i$ converges almost surely iff it converges in probability; see e.g. this question
$endgroup$
– saz
Dec 12 '18 at 15:59
add a comment |
1
1
1
$begingroup$
Let $(X_i)$ be sequence of independent random variables with $|X_i| < 1$ almost-surely and define $M_n :=sum_{i=1}^{n}X_i$. Suppose it is known that $M_n$ converges in probability to some random variable $M$, where $M < infty$ almost-surely. Does it follow that $sum_{i=1}^{infty}mathbb{E}[X_i] < infty$?
probability probability-theory convergence random-variables
asked Dec 11 '18 at 23:27
jesterIIjesterII
1,21321326
$endgroup$
Let $(X_i)$ be sequence of independent random variables with $|X_i| < 1$ almost-surely and define $M_n :=sum_{i=1}^{n}X_i$. Suppose it is known that $M_n$ converges in probability to some random variable $M$, where $M < infty$ almost-surely. Does it follow that $sum_{i=1}^{infty}mathbb{E}[X_i] < infty$?
probability probability-theory convergence random-variables
probability probability-theory convergence random-variables
asked Dec 11 '18 at 23:27
jesterIIjesterII
1,21321326
asked Dec 11 '18 at 23:27
jesterIIjesterII
1,21321326
edited Dec 12 '18 at 1:18
jesterII
asked Dec 11 '18 at 23:27
jesterIIjesterII
1,21321326
asked Dec 11 '18 at 23:27
jesterIIjesterII
1,21321326
asked Dec 11 '18 at 23:27
jesterIIjesterII
1,21321326
1,21321326
$begingroup$
Yes, the series converges... the proof which i know is somewhat technical; it uses an symmetrization argument, i.e. $Y_i := X_i-tilde{X}_i$ where $tilde{X}_i$ is an independent copy of $X_i$
$endgroup$
– saz
Dec 12 '18 at 6:36
$begingroup$
@saz Any reference or further details would be appreciated.
$endgroup$
– jesterII
Dec 12 '18 at 13:30
1
$begingroup$
Do you know Kolmogorov's three series convergence theorem? You can deduce it immediately from this result.
$endgroup$
– saz
Dec 12 '18 at 13:54
$begingroup$
@saz I've come across the three series theorem, although that applies to convergence a.s. whereas in this case the series is assumed to converge in probability. Is there an immediate way to connect the two?
$endgroup$
– jesterII
Dec 12 '18 at 15:56
1
$begingroup$
A well-known result (by Lévy) states that for independent random variables $X_i$ the series $sum_i X_i$ converges almost surely iff it converges in probability; see e.g. this question
$endgroup$
– saz
Dec 12 '18 at 15:59
add a comment |
$begingroup$
Yes, the series converges... the proof which i know is somewhat technical; it uses an symmetrization argument, i.e. $Y_i := X_i-tilde{X}_i$ where $tilde{X}_i$ is an independent copy of $X_i$
$endgroup$
– saz
Dec 12 '18 at 6:36
$begingroup$
@saz Any reference or further details would be appreciated.
$endgroup$
– jesterII
Dec 12 '18 at 13:30
1
$begingroup$
Do you know Kolmogorov's three series convergence theorem? You can deduce it immediately from this result.
$endgroup$
– saz
Dec 12 '18 at 13:54
$begingroup$
@saz I've come across the three series theorem, although that applies to convergence a.s. whereas in this case the series is assumed to converge in probability. Is there an immediate way to connect the two?
$endgroup$
– jesterII
Dec 12 '18 at 15:56
1
$begingroup$
A well-known result (by Lévy) states that for independent random variables $X_i$ the series $sum_i X_i$ converges almost surely iff it converges in probability; see e.g. this question
$endgroup$
– saz
Dec 12 '18 at 15:59
$begingroup$
Yes, the series converges... the proof which i know is somewhat technical; it uses an symmetrization argument, i.e. $Y_i := X_i-tilde{X}_i$ where $tilde{X}_i$ is an independent copy of $X_i$
$endgroup$
– saz
Dec 12 '18 at 6:36
$begingroup$
Yes, the series converges... the proof which i know is somewhat technical; it uses an symmetrization argument, i.e. $Y_i := X_i-tilde{X}_i$ where $tilde{X}_i$ is an independent copy of $X_i$
$endgroup$
– saz
Dec 12 '18 at 6:36
$begingroup$
@saz Any reference or further details would be appreciated.
$endgroup$
– jesterII
Dec 12 '18 at 13:30
$begingroup$
@saz Any reference or further details would be appreciated.
$endgroup$
– jesterII
Dec 12 '18 at 13:30
1
1
$begingroup$
Do you know Kolmogorov's three series convergence theorem? You can deduce it immediately from this result.
$endgroup$
– saz
Dec 12 '18 at 13:54
$begingroup$
Do you know Kolmogorov's three series convergence theorem? You can deduce it immediately from this result.
$endgroup$
– saz
Dec 12 '18 at 13:54
$begingroup$
@saz I've come across the three series theorem, although that applies to convergence a.s. whereas in this case the series is assumed to converge in probability. Is there an immediate way to connect the two?
$endgroup$
– jesterII
Dec 12 '18 at 15:56
$begingroup$
@saz I've come across the three series theorem, although that applies to convergence a.s. whereas in this case the series is assumed to converge in probability. Is there an immediate way to connect the two?
$endgroup$
– jesterII
Dec 12 '18 at 15:56
1
1
$begingroup$
A well-known result (by Lévy) states that for independent random variables $X_i$ the series $sum_i X_i$ converges almost surely iff it converges in probability; see e.g. this question
$endgroup$
– saz
Dec 12 '18 at 15:59
$begingroup$
A well-known result (by Lévy) states that for independent random variables $X_i$ the series $sum_i X_i$ converges almost surely iff it converges in probability; see e.g. this question
$endgroup$
– saz
Dec 12 '18 at 15:59
add a comment |
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$begingroup$
Yes, the series converges... the proof which i know is somewhat technical; it uses an symmetrization argument, i.e. $Y_i := X_i-tilde{X}_i$ where $tilde{X}_i$ is an independent copy of $X_i$
$endgroup$
– saz
Dec 12 '18 at 6:36
$begingroup$
@saz Any reference or further details would be appreciated.
$endgroup$
– jesterII
Dec 12 '18 at 13:30
1
$begingroup$
Do you know Kolmogorov's three series convergence theorem? You can deduce it immediately from this result.
$endgroup$
– saz
Dec 12 '18 at 13:54
$begingroup$
@saz I've come across the three series theorem, although that applies to convergence a.s. whereas in this case the series is assumed to converge in probability. Is there an immediate way to connect the two?
$endgroup$
– jesterII
Dec 12 '18 at 15:56
1
$begingroup$
A well-known result (by Lévy) states that for independent random variables $X_i$ the series $sum_i X_i$ converges almost surely iff it converges in probability; see e.g. this question
$endgroup$
– saz
Dec 12 '18 at 15:59