Why Is It Acceptable to Interchange Sum and Integral for Martingale Proof?

• osprey
The idea is that if you can find some other integrable function that dominates the integrand, then you can switch limits and integrals since the tails of that dominating function will be arbitrarily small.In summary, The expression M_t is the sum of an infinite sequence of terms involving Poisson processes with intensity lambda. It is known that this series converges in the L^2 sense. To show that M is a martingale, it is necessary to swap the sum and integral. To do so, the dominated convergence theorem can be used. This involves finding a dominating function that makes the integrand arbitrarily small, allowing for the exchange of limits and integrals.
osprey
Let $M_t = \sum_{i=1}^\infty \frac{1}{i} (N_t^i - \lambda t)$ where $\{ N^i \}$ is a sequence of iid Poisson processes with intensity $\lambda$. It can be shown that the series converges in the $L^2$ sense. Why is it ok to write

$\int \left ( \sum_{i=1}^\infty \frac{1}{i} (N_t^i - \lambda t) \right ) dP = \sum_{i=1}^\infty \frac{1}{i} \int (N_t^i - \lambda t) dP$?

(I need to show that M is a martingale and to do so, I would like to interchange the sum and the integral, but I cannot find an argument that seems to work...)

Thank you in advance for your help! :-)

/O

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1. What is meant by "interchanging sum and integral"?

Interchanging sum and integral refers to the process of switching the order of summation and integration in a mathematical expression. This can be done if the expression meets certain criteria, such as being absolutely convergent.

2. Why would someone want to interchange sum and integral?

Sometimes, interchanging sum and integral can make a mathematical expression easier to evaluate or manipulate. It can also help to simplify complex calculations or solve certain types of problems.

3. What are the conditions for being able to interchange sum and integral?

The expression must be absolutely convergent, meaning that the sum of the absolute values of the terms in the series is finite. Additionally, the sum and integral must both converge.

4. What is the difference between a sum and an integral?

A sum is a mathematical operation that involves adding together a series of numbers, while an integral is a mathematical operation that involves finding the area under a curve. Sums are discrete, while integrals are continuous.

5. Can any sum and integral be interchanged?

No, not all sums and integrals can be interchanged. The expression must meet the criteria mentioned earlier, and even then, the order of summation and integration may affect the final result. It is important to carefully consider the conditions and implications before interchanging sum and integral.

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