Relationships between almost sure convergence, convergence in density, and convergence in moments

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I've been trying to map out the relationships between the modes of convergence associated with random variables. However, I've had a hard time finding out online whether the statements below are true or not. I was wondering if anyone can also give me intuition as to why are true o aside from a proof or counterexample.



1) Let $X_n$ be a sequence of random variables with probability density function $f_n$. Suppose that $X_n rightarrow X$ a.s. such that $X$ has density $f$. Does it follow that $f_n rightarrow f$ in some sense? Perhaps pointwise, in $L^1$, etc.?



2) Let $X_n$ be a sequence of random variables with density $f_n$. Suppose that $f_n rightarrow f$ almost everywhere and that $int f ,dx = 1$. Let $X$ be a random variable whose density is $f$. Do the moments of $X_n$ converge to the moments of $X$?




real-analysis probability probability-theory




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    I've been trying to map out the relationships between the modes of convergence associated with random variables. However, I've had a hard time finding out online whether the statements below are true or not. I was wondering if anyone can also give me intuition as to why are true o aside from a proof or counterexample.



    1) Let $X_n$ be a sequence of random variables with probability density function $f_n$. Suppose that $X_n rightarrow X$ a.s. such that $X$ has density $f$. Does it follow that $f_n rightarrow f$ in some sense? Perhaps pointwise, in $L^1$, etc.?



    2) Let $X_n$ be a sequence of random variables with density $f_n$. Suppose that $f_n rightarrow f$ almost everywhere and that $int f ,dx = 1$. Let $X$ be a random variable whose density is $f$. Do the moments of $X_n$ converge to the moments of $X$?




    real-analysis probability probability-theory




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      I've been trying to map out the relationships between the modes of convergence associated with random variables. However, I've had a hard time finding out online whether the statements below are true or not. I was wondering if anyone can also give me intuition as to why are true o aside from a proof or counterexample.



      1) Let $X_n$ be a sequence of random variables with probability density function $f_n$. Suppose that $X_n rightarrow X$ a.s. such that $X$ has density $f$. Does it follow that $f_n rightarrow f$ in some sense? Perhaps pointwise, in $L^1$, etc.?



      2) Let $X_n$ be a sequence of random variables with density $f_n$. Suppose that $f_n rightarrow f$ almost everywhere and that $int f ,dx = 1$. Let $X$ be a random variable whose density is $f$. Do the moments of $X_n$ converge to the moments of $X$?




      real-analysis probability probability-theory




      share|cite|improve this question












      I've been trying to map out the relationships between the modes of convergence associated with random variables. However, I've had a hard time finding out online whether the statements below are true or not. I was wondering if anyone can also give me intuition as to why are true o aside from a proof or counterexample.



      1) Let $X_n$ be a sequence of random variables with probability density function $f_n$. Suppose that $X_n rightarrow X$ a.s. such that $X$ has density $f$. Does it follow that $f_n rightarrow f$ in some sense? Perhaps pointwise, in $L^1$, etc.?



      2) Let $X_n$ be a sequence of random variables with density $f_n$. Suppose that $f_n rightarrow f$ almost everywhere and that $int f ,dx = 1$. Let $X$ be a random variable whose density is $f$. Do the moments of $X_n$ converge to the moments of $X$?





      real-analysis probability probability-theory





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      asked Aug 19 at 4:28









      Tomas Jorovic

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