Convergence in law of a stopped process to standard normal

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Let $S_T$ be a cadlag process on $[l,u]$, where $0 < l < u < 1$, such that
$$
S_T stackreldlongrightarrow S
$$
where $S(t) equiv fracW(t)sqrtt$ and $W$ is a Wiener process. It follows that at any $t$, the asymptotic limit of $S_T(t)$ is standard normal.



Now I am wondering if the asymptotic limit of $S_T(tau_T)$ is still standard normal, where $tau_T$ is a random variable taking values in $(l,u)$ and independent of the processes. I guess the answer is yes, and here is my attempt: Let $F_T$ be the cumulative distribution function of $tau_T$. By independence,
$$
mathbbE[exp(i theta S_T(tau_T))] = int_l^u mathbbE[exp(i theta S_T(t))] dF_T(t)
$$
The integrand converges pointwise to $exp(-frac12 theta^2)$. Thus as long as there is a suitable convergence theorem we can conclude the proof.



My question is then: is there such a suitable convergence theorem that I can use? Or maybe something is wrong with my reasoning? I would greatly appreciate, if there is any error, that you point out what additional assumptions are required to establish the desired conclusion.



Many thanks!




probability stochastic-processes




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    Let $S_T$ be a cadlag process on $[l,u]$, where $0 < l < u < 1$, such that
    $$
    S_T stackreldlongrightarrow S
    $$
    where $S(t) equiv fracW(t)sqrtt$ and $W$ is a Wiener process. It follows that at any $t$, the asymptotic limit of $S_T(t)$ is standard normal.



    Now I am wondering if the asymptotic limit of $S_T(tau_T)$ is still standard normal, where $tau_T$ is a random variable taking values in $(l,u)$ and independent of the processes. I guess the answer is yes, and here is my attempt: Let $F_T$ be the cumulative distribution function of $tau_T$. By independence,
    $$
    mathbbE[exp(i theta S_T(tau_T))] = int_l^u mathbbE[exp(i theta S_T(t))] dF_T(t)
    $$
    The integrand converges pointwise to $exp(-frac12 theta^2)$. Thus as long as there is a suitable convergence theorem we can conclude the proof.



    My question is then: is there such a suitable convergence theorem that I can use? Or maybe something is wrong with my reasoning? I would greatly appreciate, if there is any error, that you point out what additional assumptions are required to establish the desired conclusion.



    Many thanks!




    probability stochastic-processes




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      Let $S_T$ be a cadlag process on $[l,u]$, where $0 < l < u < 1$, such that
      $$
      S_T stackreldlongrightarrow S
      $$
      where $S(t) equiv fracW(t)sqrtt$ and $W$ is a Wiener process. It follows that at any $t$, the asymptotic limit of $S_T(t)$ is standard normal.



      Now I am wondering if the asymptotic limit of $S_T(tau_T)$ is still standard normal, where $tau_T$ is a random variable taking values in $(l,u)$ and independent of the processes. I guess the answer is yes, and here is my attempt: Let $F_T$ be the cumulative distribution function of $tau_T$. By independence,
      $$
      mathbbE[exp(i theta S_T(tau_T))] = int_l^u mathbbE[exp(i theta S_T(t))] dF_T(t)
      $$
      The integrand converges pointwise to $exp(-frac12 theta^2)$. Thus as long as there is a suitable convergence theorem we can conclude the proof.



      My question is then: is there such a suitable convergence theorem that I can use? Or maybe something is wrong with my reasoning? I would greatly appreciate, if there is any error, that you point out what additional assumptions are required to establish the desired conclusion.



      Many thanks!




      probability stochastic-processes




      share|cite|improve this question











      Let $S_T$ be a cadlag process on $[l,u]$, where $0 < l < u < 1$, such that
      $$
      S_T stackreldlongrightarrow S
      $$
      where $S(t) equiv fracW(t)sqrtt$ and $W$ is a Wiener process. It follows that at any $t$, the asymptotic limit of $S_T(t)$ is standard normal.



      Now I am wondering if the asymptotic limit of $S_T(tau_T)$ is still standard normal, where $tau_T$ is a random variable taking values in $(l,u)$ and independent of the processes. I guess the answer is yes, and here is my attempt: Let $F_T$ be the cumulative distribution function of $tau_T$. By independence,
      $$
      mathbbE[exp(i theta S_T(tau_T))] = int_l^u mathbbE[exp(i theta S_T(t))] dF_T(t)
      $$
      The integrand converges pointwise to $exp(-frac12 theta^2)$. Thus as long as there is a suitable convergence theorem we can conclude the proof.



      My question is then: is there such a suitable convergence theorem that I can use? Or maybe something is wrong with my reasoning? I would greatly appreciate, if there is any error, that you point out what additional assumptions are required to establish the desired conclusion.



      Many thanks!





      probability stochastic-processes





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      asked Aug 2 at 13:34









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