Let $\{Y(t):t\ge 0\}$ be a symmetric 1-stable Lévy process. Then $Y$ is a càdlàg process with $E[e^{iuY(t)}]=e^{-t|u|}$. A Levy process is a semimartingale, so we may define the usual stochastic integral with respect to $Y$. Consequently, if $f$ is continuous, then the stochastic integral is the limit in probability of left-endpoint Riemann sums. For example, $$ \sum_{j=1}^{n} f(t_{j-1})(Y(t_j) - Y(t_{j-1})) \to \int_0^1 f(s)\\,dY(s), $$ in probability as $n\to\infty$, where $t_j=j/n$.

Since a Lévy process has stationary, independent increments, the sum on the left is equal in law to $$ \frac1n\sum_{j=1}^{n} f(j/n)X_j, $$ where $\{X_j\}$ is an iid sequence of standard Cauchy random variables, i.e. $E[e^{iuX_1}]=e^{-|u|}$ and $X_1$ has density $1/\pi(1+x^2)$.

Protter is a standard reference on general stochastic integration, which includes discussions of Lévy processes.

Let $\{Y(t):t\ge 0\}$ be a symmetric 1-stable Lévy process. Then $Y$ is a càdlàg process with $E[e^{iuY(t)}]=e^{-t|u|}$. A Levy process is a semimartingale, so we may define the usual stochastic integral with respect to $Y$. Consequently, if $f$ is continuous, then the stochastic integral is the limit in probability of left-endpoint Riemann sums. For example, $$ \sum_{j=1}^{n} f(t_{j-1})(Y(t_j) - Y(t_{j-1})) \to \int_0^1 f(s)\,dY(s), $$ in probability as $n\to\infty$, where $t_j=j/n$.

Since a Lévy process has stationary, independent increments, the sum on the left is equal in law to $$ \frac1n\sum_{j=1}^{n} f(j/n)X_j, $$ where $\{X_j\}$ is an iid sequence of standard Cauchy random variables, i.e. $E[e^{iuX_1}]=e^{-|u|}$ and $X_1$ has density $1/\pi(1+x^2)$.

Protter is a standard reference on general stochastic integration, which includes discussions of Lévy processes.

Let $\{Y(t):t\ge 0\}$ be a symmetric 1-stable Lévy process. Then $Y$ is a càdlàg process with $E[e^{iuY(t)}]=e^{-t|u|}$. A Levy process is a semimartingale, so we may define the usual stochastic integral with respect to $Y$. Consequently, if $f$ is continuous (or even just left-continuous with right-hand limits), then the stochastic integral is the limit in probability of left-endpoint Riemann sums. For example, $$ \sum_{j=1}^{n} f(t_{j-1})(Y(t_j) - Y(t_{j-1})) \to \int_0^1 f(s)\\,dY(s), $$ uniformly on compacts in probability as $n\to\infty$, where $t_j=j/n$.

Since a Lévy process has stationary, independent increments, the sum on the left is equal in law to $$ \frac1n\sum_{j=1}^{n} f(j/n)X_j, $$ where $\{X_j\}$ is an iid sequence of standard Cauchy random variables (i.e. $E[e^{iuX_1}]=e^{-|u|}$ and $X_1$ has density $1/\pi(1+x^2)$.

Protter is a standard reference on general stochastic integration, which includes discussions of Lévy processes.

Let $\{Y(t):t\ge 0\}$ be a symmetric 1-stable Lévy process. Then $Y$ is a càdlàg process with $E[e^{iuY(t)}]=e^{-t|u|}$. A Levy process is a semimartingale, so we may define the usual stochastic integral with respect to $Y$. Consequently, if $f$ is continuous, then the stochastic integral is the limit in probability of left-endpoint Riemann sums. For example, $$ \sum_{j=1}^{n} f(t_{j-1})(Y(t_j) - Y(t_{j-1})) \to \int_0^1 f(s)\\,dY(s), $$ in probability as $n\to\infty$, where $t_j=j/n$.

Since a Lévy process has stationary, independent increments, the sum on the left is equal in law to $$ \frac1n\sum_{j=1}^{n} f(j/n)X_j, $$ where $\{X_j\}$ is an iid sequence of standard Cauchy random variables, i.e. $E[e^{iuX_1}]=e^{-|u|}$ and $X_1$ has density $1/\pi(1+x^2)$.

Protter is a standard reference on general stochastic integration, which includes discussions of Lévy processes.