Convergence of stochastic process divided by floor.

Question

Assume that $N(t)$ is a stochastic process with positive and integer values. We know that:

$\frac{N(t)}{t} \xrightarrow{\text{P}} c, \hspace{0.5cm} t \rightarrow \infty$

where $c$ is positive and integer valued random variable. I would like to prove that: $\frac{[c t]}{N(t)} \xrightarrow{\text{P}} 1$ in a formal way, because it looks intuitively.

Is this attempt even close to the answer?

\begin{align} P \left( \left| \frac{[c t]}{N(t)} - 1 \right| > \epsilon \right) &= P \left( [c t] - N(t) > \epsilon N(t) \right) + P \left( [c t] - N(t) < - \epsilon N(t) \right) \\&\leq P \left( ct + 1 - N(t) > \epsilon N(t) \right) + P \left( ct - 1 - N(t) < -\epsilon N(t) \right)\\ & = P \left( ct - N(t) > \epsilon N(t) - 1 \right) + P \left( ct - N(t) < \epsilon N(t) + 1 \right). \end{align}

And what can I do now for obtaining the proof?

Answer 1

Maybe it could be simpler to do a reasoning with sequences. It suffices to show that for each sequence $\left(t_n\right)_{n\geqslant 1}$ such that $\lim_{n\to +\infty}t_n=+\infty$, the sequence $\left(Y_n\right)_{n\geqslant 1}$ defined by $$ Y_n:=\frac{\left[ct_n\right]}{N\left(t_n\right)} $$ converges to $1$ in probability. We can use the following facts.

1. If a sequence of positive random variables $\left(Z_n\right)$ converges to $1$ in probability, so does the sequence $\left(1/Z_n\right)$.
2. If the sequences of positive random variables $\left(Z_n\right)$ and $\left(Z'_n\right)$ converges to $1$ in probability, so does $\left(Z_nZ'_n\right)$.
3. $\left(\left[ct_n\right]/\left(ct_n\right)\right)_{n\geqslant 1}$ converges to $1$ in probability.