Limiting a sequence of moment generating functions
I was trying to solve the following problem:
Let ${X_n}_{n=1}^{infty}$ be a sequence of independent random variables with the probability mass function $P{X_n = pm1 } = frac{1}{2}$, $n in mathbb{N}$. Let $Z_n=sum_{j=1}^{n}{X_j/2^j}$. Show that $Z_n xrightarrow{L} Z$, where $Z sim U[-1, 1]$.
(From An Introduction to Probability and Statistics V.K. Rohatgi & A. K. Md. Saleh, (c) 2015, Problems 7.5, Page 320)
$xrightarrow{L}$ means convergence in law (or in distribution), and $U[-1, 1]$ is the uniform distribution on the interval $[-1, 1]$.
My approach was the following:
We need to show that $limlimits_{nrightarrowinfty} M_{Z_n}(t) = M_{Z}(t) = frac{e^{1 times t} - e^{-1 times t}}{t times (1 - (-1))} = frac{e^t - e^{-t}}{2t}$.
$$M_{Z_n}(t) = E_{Z_n}left(e^{tZ_n}right) = Eleft(e^{tsum_{j = 1}^{n}{frac{X_j}{2^j}}}right) = Eleft(prod_{j=1}^{n}{e^{tfrac{X_j}{2^j}}} right) = prod_{j=1}^{n}{E_{X_j}left(e^{tfrac{X_j}{2^j}} right)}$$
$$ E_{X_j}left( e^{t frac{X_j}{2^j}} right) = e^{t times frac{-1}{2^j}} times frac{1}{2} + e^{t times frac{1}{2^j}} times frac{1}{2} = frac{1}{2} left( e^{frac{t}{2^j}} + e^{frac{-t}{2^j}} right) $$
Hence,
$$
M_{Z_n}(t) = prod_{j = 1}^{n}{frac{1}{2} left( e^{frac{t}{2^j}} + e^{frac{-t}{2^j}} right)}
$$
I cannot see how this sequence of functions converges to the required moment generating function of $U[-1,1]$.
I had many attempts, for instance using the power series representation of $e^x$ and limiting approximations, but failed in them all. After that I started thinking that perhaps I'm missing knowledge of some theorem.
Any idea how to proceed?
probability-theory real-analysis analysis probability-distributions
asked Dec 24 '18 at 14:39
Noor AlYaqeenNoor AlYaqeen
346
migrated from mathoverflow.net 2 days ago
This question came from our site for professional mathematicians.
add a comment |
I was trying to solve the following problem:
Let ${X_n}_{n=1}^{infty}$ be a sequence of independent random variables with the probability mass function $P{X_n = pm1 } = frac{1}{2}$, $n in mathbb{N}$. Let $Z_n=sum_{j=1}^{n}{X_j/2^j}$. Show that $Z_n xrightarrow{L} Z$, where $Z sim U[-1, 1]$.
(From An Introduction to Probability and Statistics V.K. Rohatgi & A. K. Md. Saleh, (c) 2015, Problems 7.5, Page 320)
$xrightarrow{L}$ means convergence in law (or in distribution), and $U[-1, 1]$ is the uniform distribution on the interval $[-1, 1]$.
My approach was the following:
We need to show that $limlimits_{nrightarrowinfty} M_{Z_n}(t) = M_{Z}(t) = frac{e^{1 times t} - e^{-1 times t}}{t times (1 - (-1))} = frac{e^t - e^{-t}}{2t}$.
$$M_{Z_n}(t) = E_{Z_n}left(e^{tZ_n}right) = Eleft(e^{tsum_{j = 1}^{n}{frac{X_j}{2^j}}}right) = Eleft(prod_{j=1}^{n}{e^{tfrac{X_j}{2^j}}} right) = prod_{j=1}^{n}{E_{X_j}left(e^{tfrac{X_j}{2^j}} right)}$$
$$ E_{X_j}left( e^{t frac{X_j}{2^j}} right) = e^{t times frac{-1}{2^j}} times frac{1}{2} + e^{t times frac{1}{2^j}} times frac{1}{2} = frac{1}{2} left( e^{frac{t}{2^j}} + e^{frac{-t}{2^j}} right) $$
Hence,
$$
M_{Z_n}(t) = prod_{j = 1}^{n}{frac{1}{2} left( e^{frac{t}{2^j}} + e^{frac{-t}{2^j}} right)}
$$
I cannot see how this sequence of functions converges to the required moment generating function of $U[-1,1]$.
I had many attempts, for instance using the power series representation of $e^x$ and limiting approximations, but failed in them all. After that I started thinking that perhaps I'm missing knowledge of some theorem.
Any idea how to proceed?
probability-theory real-analysis analysis probability-distributions
asked Dec 24 '18 at 14:39
Noor AlYaqeenNoor AlYaqeen
346
migrated from mathoverflow.net 2 days ago
This question came from our site for professional mathematicians.
add a comment |
I was trying to solve the following problem:
Let ${X_n}_{n=1}^{infty}$ be a sequence of independent random variables with the probability mass function $P{X_n = pm1 } = frac{1}{2}$, $n in mathbb{N}$. Let $Z_n=sum_{j=1}^{n}{X_j/2^j}$. Show that $Z_n xrightarrow{L} Z$, where $Z sim U[-1, 1]$.
(From An Introduction to Probability and Statistics V.K. Rohatgi & A. K. Md. Saleh, (c) 2015, Problems 7.5, Page 320)
$xrightarrow{L}$ means convergence in law (or in distribution), and $U[-1, 1]$ is the uniform distribution on the interval $[-1, 1]$.
My approach was the following:
We need to show that $limlimits_{nrightarrowinfty} M_{Z_n}(t) = M_{Z}(t) = frac{e^{1 times t} - e^{-1 times t}}{t times (1 - (-1))} = frac{e^t - e^{-t}}{2t}$.
$$M_{Z_n}(t) = E_{Z_n}left(e^{tZ_n}right) = Eleft(e^{tsum_{j = 1}^{n}{frac{X_j}{2^j}}}right) = Eleft(prod_{j=1}^{n}{e^{tfrac{X_j}{2^j}}} right) = prod_{j=1}^{n}{E_{X_j}left(e^{tfrac{X_j}{2^j}} right)}$$
$$ E_{X_j}left( e^{t frac{X_j}{2^j}} right) = e^{t times frac{-1}{2^j}} times frac{1}{2} + e^{t times frac{1}{2^j}} times frac{1}{2} = frac{1}{2} left( e^{frac{t}{2^j}} + e^{frac{-t}{2^j}} right) $$
Hence,
$$
M_{Z_n}(t) = prod_{j = 1}^{n}{frac{1}{2} left( e^{frac{t}{2^j}} + e^{frac{-t}{2^j}} right)}
$$
I cannot see how this sequence of functions converges to the required moment generating function of $U[-1,1]$.
I had many attempts, for instance using the power series representation of $e^x$ and limiting approximations, but failed in them all. After that I started thinking that perhaps I'm missing knowledge of some theorem.
Any idea how to proceed?
probability-theory real-analysis analysis probability-distributions
asked Dec 24 '18 at 14:39
Noor AlYaqeenNoor AlYaqeen
346
I was trying to solve the following problem:
Let ${X_n}_{n=1}^{infty}$ be a sequence of independent random variables with the probability mass function $P{X_n = pm1 } = frac{1}{2}$, $n in mathbb{N}$. Let $Z_n=sum_{j=1}^{n}{X_j/2^j}$. Show that $Z_n xrightarrow{L} Z$, where $Z sim U[-1, 1]$.
(From An Introduction to Probability and Statistics V.K. Rohatgi & A. K. Md. Saleh, (c) 2015, Problems 7.5, Page 320)
$xrightarrow{L}$ means convergence in law (or in distribution), and $U[-1, 1]$ is the uniform distribution on the interval $[-1, 1]$.
My approach was the following:
We need to show that $limlimits_{nrightarrowinfty} M_{Z_n}(t) = M_{Z}(t) = frac{e^{1 times t} - e^{-1 times t}}{t times (1 - (-1))} = frac{e^t - e^{-t}}{2t}$.
$$M_{Z_n}(t) = E_{Z_n}left(e^{tZ_n}right) = Eleft(e^{tsum_{j = 1}^{n}{frac{X_j}{2^j}}}right) = Eleft(prod_{j=1}^{n}{e^{tfrac{X_j}{2^j}}} right) = prod_{j=1}^{n}{E_{X_j}left(e^{tfrac{X_j}{2^j}} right)}$$
$$ E_{X_j}left( e^{t frac{X_j}{2^j}} right) = e^{t times frac{-1}{2^j}} times frac{1}{2} + e^{t times frac{1}{2^j}} times frac{1}{2} = frac{1}{2} left( e^{frac{t}{2^j}} + e^{frac{-t}{2^j}} right) $$
Hence,
$$
M_{Z_n}(t) = prod_{j = 1}^{n}{frac{1}{2} left( e^{frac{t}{2^j}} + e^{frac{-t}{2^j}} right)}
$$
I cannot see how this sequence of functions converges to the required moment generating function of $U[-1,1]$.
I had many attempts, for instance using the power series representation of $e^x$ and limiting approximations, but failed in them all. After that I started thinking that perhaps I'm missing knowledge of some theorem.
Any idea how to proceed?
probability-theory real-analysis analysis probability-distributions
probability-theory real-analysis analysis probability-distributions
asked Dec 24 '18 at 14:39
Noor AlYaqeenNoor AlYaqeen
346
asked Dec 24 '18 at 14:39
Noor AlYaqeenNoor AlYaqeen
346
asked Dec 24 '18 at 14:39
Noor AlYaqeenNoor AlYaqeen
346
asked Dec 24 '18 at 14:39
Noor AlYaqeenNoor AlYaqeen
346
asked Dec 24 '18 at 14:39
Noor AlYaqeenNoor AlYaqeen
346
346
migrated from mathoverflow.net 2 days ago
This question came from our site for professional mathematicians.
migrated from mathoverflow.net 2 days ago
This question came from our site for professional mathematicians.
add a comment |
add a comment |
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