Law of large numbers with continuous dependency

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Suppose we have an i.i.d. sequence of random variables $X_n sim X$, and a sequence $Y_n to y_0$ in probability, where $y_0$ is a constant. Suppose also that $f : mathbbR^2 to mathbbR$ is continuous.



First question: Is it the case that



$$frac1n sum_i=1^n f(X_i, Y_n) to mathbbE[f(X, y_0)]$$



in probability? I suspect the answer is no but I am having trouble finding a counterexample.



The result does hold provided we put additional assumptions on $f$. For instance, either



(1) that $f(x, cdot)$ is $L(x)$-Lipschitz for each $x$, with $mathbbE[L(X)] < infty$;



or



(2) that $f(cdot, y) to f(cdot, y_0)$ in the supremum norm whenever $y to y_0$.



Second question: is there a relationship between these two conditions? That is, is one strictly weaker than the other? For instance, if (1) holds with $L(x)$ bounded then it is clear that (2) holds. But how about more generally?






probability-theory probability-limit-theorems







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  • I would search for an example where $Ef(X,y)$ is not continuous in $y$ at $y_0$, or for examples of non-uniform integrability.
    – kimchi lover
    Sep 3 at 14:53














up vote
1
down vote

favorite












Suppose we have an i.i.d. sequence of random variables $X_n sim X$, and a sequence $Y_n to y_0$ in probability, where $y_0$ is a constant. Suppose also that $f : mathbbR^2 to mathbbR$ is continuous.



First question: Is it the case that



$$frac1n sum_i=1^n f(X_i, Y_n) to mathbbE[f(X, y_0)]$$



in probability? I suspect the answer is no but I am having trouble finding a counterexample.



The result does hold provided we put additional assumptions on $f$. For instance, either



(1) that $f(x, cdot)$ is $L(x)$-Lipschitz for each $x$, with $mathbbE[L(X)] < infty$;



or



(2) that $f(cdot, y) to f(cdot, y_0)$ in the supremum norm whenever $y to y_0$.



Second question: is there a relationship between these two conditions? That is, is one strictly weaker than the other? For instance, if (1) holds with $L(x)$ bounded then it is clear that (2) holds. But how about more generally?






probability-theory probability-limit-theorems







share|cite|improve this question





















  • I would search for an example where $Ef(X,y)$ is not continuous in $y$ at $y_0$, or for examples of non-uniform integrability.
    – kimchi lover
    Sep 3 at 14:53












up vote
1
down vote

favorite









up vote
1
down vote

favorite











Suppose we have an i.i.d. sequence of random variables $X_n sim X$, and a sequence $Y_n to y_0$ in probability, where $y_0$ is a constant. Suppose also that $f : mathbbR^2 to mathbbR$ is continuous.



First question: Is it the case that



$$frac1n sum_i=1^n f(X_i, Y_n) to mathbbE[f(X, y_0)]$$



in probability? I suspect the answer is no but I am having trouble finding a counterexample.



The result does hold provided we put additional assumptions on $f$. For instance, either



(1) that $f(x, cdot)$ is $L(x)$-Lipschitz for each $x$, with $mathbbE[L(X)] < infty$;



or



(2) that $f(cdot, y) to f(cdot, y_0)$ in the supremum norm whenever $y to y_0$.



Second question: is there a relationship between these two conditions? That is, is one strictly weaker than the other? For instance, if (1) holds with $L(x)$ bounded then it is clear that (2) holds. But how about more generally?






probability-theory probability-limit-theorems







share|cite|improve this question













Suppose we have an i.i.d. sequence of random variables $X_n sim X$, and a sequence $Y_n to y_0$ in probability, where $y_0$ is a constant. Suppose also that $f : mathbbR^2 to mathbbR$ is continuous.



First question: Is it the case that



$$frac1n sum_i=1^n f(X_i, Y_n) to mathbbE[f(X, y_0)]$$



in probability? I suspect the answer is no but I am having trouble finding a counterexample.



The result does hold provided we put additional assumptions on $f$. For instance, either



(1) that $f(x, cdot)$ is $L(x)$-Lipschitz for each $x$, with $mathbbE[L(X)] < infty$;



or



(2) that $f(cdot, y) to f(cdot, y_0)$ in the supremum norm whenever $y to y_0$.



Second question: is there a relationship between these two conditions? That is, is one strictly weaker than the other? For instance, if (1) holds with $L(x)$ bounded then it is clear that (2) holds. But how about more generally?






probability-theory probability-limit-theorems




probability-theory probability-limit-theorems






share|cite|improve this question













share|cite|improve this question











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asked Sep 3 at 9:57









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  • I would search for an example where $Ef(X,y)$ is not continuous in $y$ at $y_0$, or for examples of non-uniform integrability.
    – kimchi lover
    Sep 3 at 14:53
















  • I would search for an example where $Ef(X,y)$ is not continuous in $y$ at $y_0$, or for examples of non-uniform integrability.
    – kimchi lover
    Sep 3 at 14:53















I would search for an example where $Ef(X,y)$ is not continuous in $y$ at $y_0$, or for examples of non-uniform integrability.
– kimchi lover
Sep 3 at 14:53




I would search for an example where $Ef(X,y)$ is not continuous in $y$ at $y_0$, or for examples of non-uniform integrability.
– kimchi lover
Sep 3 at 14:53















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