Stochastic Processes and Financial Mathematics
(part one)
6.3 Exercises on Chapter 6
On convergence of random variables
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6.1 Let \((X_n)\) be a sequence of independent random variables such that
\[ X_n= \begin {cases} 2^{-n} & \text { with probability }\frac {1}{2}\\ 0 & \text { with probability }\frac {1}{2}.\\ \end {cases} \]
Show that \(X_n\to 0\) in \(L^1\) and almost surely. Deduce that also \(X_n\to 0\) in probability and in distribution.
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6.2 Let \(X_n,X\) be random variables.
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(a) Suppose that \(X_n\stackrel {L^1}{\to } X\) as \(n\to \infty \). Show that \(\E [X_n]\to \E [X]\).
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(b) Give an example where \(\E [X_n]\to \E [X]\) but \(X_n\) does not converge to \(X\) in \(L^1\).
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6.3 Let \(U\) be a random variable such that \(\P [U=0]=\P [U=1]=\P [U=2]=\frac 13\). Let \(X_n\) and \(X\) be given by
\[ X_n= \begin {cases} 1+\frac {1}{n} & \text { if }U=0\\ 1-\frac {1}{n} & \text { if }U=1\\ 0 & \text { if }U=2, \end {cases} \quad \quad \quad \quad X= \begin {cases} 1 & \text { if }U\in \{0,1\}\\ 0 & \text { if }U=2. \end {cases} \]
Show that \(X_n\to X\) both almost surely and in \(L^1\). Deduce that also \(X_n\to X\) in probability and in distribution.
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6.4 Let \(X_1\) be a random variable with distribution given by \(\P [X_1=1]=\P [X_1=0]=\frac 12\). Set \(X_n=X_1\) for all \(n\geq 2\). Set \(Y=1-X_1\). Show that \(X_n\to Y\) in distribution, but not in probability.
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6.5 Let \((X_n)\) be the sequence of random variables from 6.1. Define \(Y_n=X_1+X_2+\ldots +X_n\).
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(a) Show that, for all \(\omega \in \Omega \), the sequence \(Y_n(\omega )\) is increasing and bounded.
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(b) Deduce that there exists a random variable \(Y\) such that \(Y_n\stackrel {a.s.}{\to }Y\).
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(c) Write down the distribution of \(Y_1,Y_2\) and \(Y_3\).
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(d) Suggest why we might guess that \(Y\) has a uniform distribution on \([0,1]\).
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(e) Prove that \(Y_n\) has a uniform distribution on \(\{k2^{-n}\-k=0,1,\ldots ,2^n-1\}\).
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(f) Prove that \(Y\) has a uniform distribution on \([0,1]\).
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On the monotone convergence theorem
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6.6 Let \(Y_n\) be a sequence of random variables such that \(Y_{n+1}\leq Y_n\leq 0\), almost surely, for all \(n\). Show that there exists a random variable \(Y\) such that \(Y_n\stackrel {a.s.}{\to }Y\) and \(\E [Y_n]\to \E [Y]\).
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6.7 Let \(X\) be a random variable such that \(X\geq 0\) and \(\P [X<\infty ]=1\). Define
\[ X_n= \begin {cases} X & \text { if }X\leq n\\ 0 & \text {otherwise.} \end {cases} \]
Equivalently, \(X_n=X\1\{X\leq n\}\). Show that \(\E [X_n]\to \E [X]\) as \(n\to \infty \).
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6.8 Let \((X_n)\) be a sequence of random variables.
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(a) Explain briefly why \(\E \big [\sum _{i=1}^n X_i\big ]=\sum _{i=1}^n \E [X_i]\) follows from the linearity of \(\E \), for \(n\in \N \). Explain briefly why linearity alone does not allow us to deduce the same equation with \(n=\infty \).
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(b) Suppose that \(X_n\geq 0\) almost surely, for all \(n\). Show that
\(\seteqnumber{0}{6.}{0}\)\begin{equation} \label {eq:mct_fubini} \E \l [\sum _{i=1}^\infty X_i\r ]=\sum _{i=1}^\infty \E [X_i]. \end{equation}
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(c) Suppose, instead, that the \(X_i\) are independent and that \(\P [X_i=1]=\P [X_i=-1]=\frac 12\). Explain briefly why (6.1) fails to hold, in this case.
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Challenge questions
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6.6 Let \((X_n)\) be a sequence of random variables, and let \(X\) and \(Y\) be random variables.
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(a) Show that if \(X_n\stackrel {d}{\to }X\) and \(X_n\stackrel {d}{\to }Y\) then \(X\) and \(Y\) have the same distribution.
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(b) Show that if \(X_n\stackrel {\P }{\to }X\) and \(X_n\stackrel {\P }{\to }Y\) then \(X=Y\) almost surely.
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6.7 Let \((X_n)\) be a sequence of independent random variables such that \(\P [X_n=1]=\P [X_n=0]=\frac 12\). Show that \((X_n)\) does not converge in probability and deduce that \((X_n)\) also does not converge in \(L^1\), or almost surely. Does \(X_n\) converge in distribution?